What Are Dependent and Independent Events in Math?

Events are independent of each other if one event doesn’t affect the probability of the other event occurring. To find out whether the events $A$ and $B$ are independent of each other or not, you have to check whether the probability of $A$ changes when you already know that $B$ has occurred. This probability is written $P (A ∣ B)$ , and is read as “the probability of $A$ given $B$ ”, where the vertical bar between $A$ and $B$ is the sign that means “given”.

Rule

Independence

When $A$ and $B$ are independent events, we have that

P (A ∣ B) = P (A)

and

P (A \cap B) = P (A) \cdot P (B)

Rule

Dependence

If $A$ and $B$ are events that depend on each other, we have that

P (A ∣ B) \neq P (A)

and

P (A \cap B) = P (A ∣ B) \cdot P (B)

The set $A \cap B$ in the boxes above is called the intersection of $A$ and $B$ , and the probability $P (A \cap B)$ is found through the chain rule.

Example 1

Decide whether the two events are independent when you know that

\begin{array}{l} PIn this case you can simply check whether P (A ∣ B) = P (A), where A = lung cancer and B = smoking . If it is, the events are independent of each other: \begin{array}{l} P (lung cancer ∣ smoking) \\ = 0.000 06 \\ \neq 0.000 02 \\ = P (lung cancer) \end{array} \begin{array}{l} P (lung cancer ∣ smoking) & = 0.000 06 \\ \neq 0.000 02 \\ = P (lung cancer) \end{array} Because you got an inequality instead of an equation, you know the events are dependent on each other, which would mean that smoking and getting lung cancer are events that are dependent on each other. Note! The numbers in this example are made up!Want to know more? Sign Up It's free!Previous entry What Are Venn Diagrams in Math? Next entry What Is a Conditional Probability?Go backHome All Math Games Statistics ChatAbout Us About House of Math Employees Career Media Lectures Blog Contact support@houseofmath.com Booking of private tutoring +47 22 150 300 Address Bygdøy allé 23
            0262 Oslo
            Norway FAQ Links WeTakeAction Math Magic for Ukraine Tutoring Video Crash Courses Junior Math Math Essentials Encyclopedia Language Norsk Українська Terms Privacy{"props":{"pageProps":{"campaign":false,"topic":{"type":"encyclopedia","language":"en","metaDescription":"Learn about dependent and independent events. Events are independent of each other if one event doesn't affect the probability of the other event occurring.","htmlContent":{"html":" \u003c/p\u003e\u003cp class=\"indent\" \u003e     Events are \u003cspan  class=\"ec-lmri-12x-x-144\"\u003eindependent \u003c/span\u003eof each other if one event doesn’t affect the \u003ca href=\"/encyclopedia/statistics-and-probability/probability-and-combinatorics/what-is-probability\"\u003eprobability\u003c/a\u003e of the other event occurring. To find out whether the events \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eA\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/math\u003e are independent of each other or not, you have to check whether the probability of \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eA\u003c/mi\u003e\u003c/math\u003e changes when you already know that \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/math\u003e has occurred. This probability is written \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003eA\u003c/mi\u003e\u003cmo  class=\"MathClass-rel\"\u003e∣\u003c/mo\u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/math\u003e, and is read as “the probability of \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eA\u003c/mi\u003e\u003c/math\u003e given \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/math\u003e”, where the vertical bar between \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eA\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/math\u003e is the sign that means “given”. \u003c/p\u003e                                                                                                                                                                                                                                               \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-116\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Rule\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"rule-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eIndependence\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003eWhen \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eA\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/math\u003e are \u003cspan  class=\"ec-lmri-12x-x-144\"\u003eindependent \u003c/span\u003eevents, we have that \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                                 \u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003eA\u003c/mi\u003e\u003cmo  class=\"MathClass-rel\"\u003e∣\u003c/mo\u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e \u003cp class=\"noindent\" \u003eand \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                            \u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003eA\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e∩\u003c/mo\u003e \u003cmi  \u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                                           \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-117\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Rule\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"rule-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eDependence\u003c/span\u003e \u003c/h3\u003e \u003c/p\u003e\u003cp class=\"noindent\" \u003eIf \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eA\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/math\u003e are events that \u003cspan  class=\"ec-lmri-12x-x-144\"\u003edepend \u003c/span\u003eon each other, we have that \u003c/p\u003e\u003ctable class=\"equation-star\"\u003e\u003ctr\u003e\u003ctd\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" class=\"equation\"\u003e                                                  \u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003eA\u003c/mi\u003e\u003cmo  class=\"MathClass-rel\"\u003e∣\u003c/mo\u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmo  class=\"MathClass-rel\"\u003e≠\u003c/mo\u003e\u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" 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form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003eA\u003c/mi\u003e\u003cmo  class=\"MathClass-rel\"\u003e∣\u003c/mo\u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003c/math\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/table\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                                      \u003cp class=\"indent\" \u003e     The set \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eA\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e∩\u003c/mo\u003e \u003cmi  \u003eB\u003c/mi\u003e\u003c/math\u003e in the boxes above is called the \u003ca href=\"/encyclopedia/statistics-and-probability/probability-and-combinatorics/basic-probability/what-is-union-and-intersection-of-sets\"\u003eintersection\u003c/a\u003e of \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eA\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/math\u003e, and the probability \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmi  \u003eA\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\"\u003e∩\u003c/mo\u003e \u003cmi  \u003eB\u003c/mi\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e\u003c/math\u003e is found through the \u003ca href=\"/encyclopedia/statistics-and-probability/probability-and-combinatorics/rules-of-probability/the-chain-rule-of-conditional-probabilities\"\u003echain rule\u003c/a type=\"punct\"\u003e. \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-118\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Example 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eDecide whether the two events are independent when you know that\u003c/span\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                        \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmstyle mathvariant=\"bold-sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003esmoking\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                      \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmstyle mathvariant=\"bold-sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003elung cancer\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"0.17em\" class=\"thinspace\"/\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmo  class=\"MathClass-punc\"\u003e,\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmstyle mathvariant=\"bold-sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003elung cancer\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmo  class=\"MathClass-rel\"\u003e∣\u003c/mo\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003esmoking\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"0.17em\" class=\"thinspace\"/\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e\u003c/p\u003e                                                                               \u003cdiv class=\"lowerbox\"\u003e                                                                                                                                                                                                                                           \u003cp class=\"noindent\" \u003eIn this case you can simply check whether \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003eA\u003c/mi\u003e\u003cmo  class=\"MathClass-rel\"\u003e∣\u003c/mo\u003e\u003cmi  \u003eB\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmi  \u003eA\u003c/mi\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/math\u003e, where \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eA\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003elung cancer\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/math\u003e and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eB\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003esmoking\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/math\u003e. If it is, the events are independent of each other:  \u003cdiv class=\"tight\"\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                                        \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                       \u003cmtd  class=\"align-even\"\u003e \u003cmphantom\u003e  \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e  \u003c/mphantom\u003e \u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003elung cancer\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmo  class=\"MathClass-rel\"\u003e∣\u003c/mo\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003esmoking\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                       \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                       \u003cmtd  class=\"align-label\"\u003e                                        \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"0.17em\" class=\"thinspace\"/\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                           \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                       \u003cmtd  class=\"align-label\"\u003e                                        \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                       \u003cmtd  class=\"align-even\"\u003e\u003cmo  class=\"MathClass-rel\"\u003e≠\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"0.17em\" class=\"thinspace\"/\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                       \u003cmtd  class=\"align-label\"\u003e                                        \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003elung cancer\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                       \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003e\u003c/div\u003e \u003cdiv class=\"wide\"\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                              \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003elung cancer\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmo  class=\"MathClass-rel\"\u003e∣\u003c/mo\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003esmoking\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"0.17em\" class=\"thinspace\"/\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-label\"\u003e                              \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                                           \u003cmtd  class=\"align-even\"\u003e\u003cmo  class=\"MathClass-rel\"\u003e≠\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"0.17em\" class=\"thinspace\"/\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                       \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-label\"\u003e                              \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                                           \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e=\u003c/mo\u003e \u003cmstyle mathvariant=\"sans-serif\"\u003e\u003cmi  \u003eP\u003c/mi\u003e\u003c/mstyle\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003elung cancer\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003e\u003c/div\u003e  Because you got an inequality instead of an equation, you know the events are dependent on each other, which would mean that smoking and getting lung cancer are events that are dependent on each other. \u003c/p\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eNote!\u003c/span\u003e\u003c/b\u003e The numbers in this example are made up! \u003c/p\u003e                                                 \u003c/div\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                                      \u003cp class=\"indent\" \u003e     "},"siblings":[{"h1":"What Is the Addition Rule of 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