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Conditional Probability & Bayes' Theorem · Question 26 of 125

Are E and F independent?

Question 26

HardOriginal question

Two fair dice are thrown. $E$ : the sum is 9. $F$ : at least one die shows a 5.

Are $E$ and $F$ independent?

No:

P(E \cap F) = \frac{1}{18}

but

P(E)\,P(F) = \frac{11}{324}

Correct answerYour answerYes: the two dice are thrown independently of each otherCorrect answerYour answerNo: they are mutually exclusive, so they cannot happen togetherCorrect answerYour answerYes:

P(E \mid F) = P(E) = \frac{1}{9}

, so knowing

F

changes nothingCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

$P(E \cap F) = \frac{2}{36} = \frac{1}{18}$ .
$P(E)\,P(F) = \frac{1}{9} \times \frac{11}{36} = \frac{11}{324} \ne \frac{1}{18}$ : not independent.

Worth noting

They share $(4, 5)$ and $(5, 4)$ , so they are not exclusive; and $P(E \mid F) = \frac{2}{11} \ne \frac{1}{9}$ .

Concept: Testing for independence
Typical timeTypical: 150 s
Your timeYours: —

The rule this testsConditional probability

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