Math Problems Vault

Problem 1.5.1

Let $A$ and $B$ be independent events; show that $A^{\complement }, B$ are independent, and deduce that $A^{\complement }, B^{\complement }$ are independent.

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Problem 1.5.2

We roll a die $n$ times. Let $A_{i j}$ be the event that the $i$ th and $j$ th rolls produce the same number. Show that the events $\left\{ A_{i j}: 1 \leq i < j \leq n\right\}$ are pairwise independent but not independent.

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Problem 1.5.3

A fair coin is tossed repeatedly. Show that the following two statements are equivalent:

(a) the outcomes of different tosses are independent,

(b) for any given finite sequence of heads and tails, the chance of this sequence occurring in the first $m$ tosses is $2^{-m}$ , where $m$ is the length of the sequence.

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Problem 1.5.4

Let $\Omega = \left\{ 1,2, \ldots , p\right\}$ where $p$ is prime, $\mathcal{F}$ be the set of all subsets of $\Omega$ , and $\mathbb {P}\left(A\right) = |A| / p$ for all $A \in \mathcal{F}$ . Show that, if $A$ and $B$ are independent events, then at least one of $A$ and $B$ is either $\varnothing$ or $\Omega$ .

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Problem 1.5.5

Show that the conditional independence of $A$ and $B$ given $C$ neither implies, nor is implied by, the independence of $A$ and $B$ . For which events $C$ is it the case that, for all $A$ and $B$ , the events $A$ and $B$ are independent if and only if they are conditionally independent given $C$ ?

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Problem 1.5.6

Safe or sorry? Some form of prophylaxis is said to be 90 per cent effective at prevention during one year's treatment. If the degrees of effectiveness in different years are independent, show that the treatment is more likely than not to fail within 7 years.

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Problem 1.5.7

(a)(b)(c)(d)

Jane has three children, each of which is equally likely to be a boy or a girl independently of the others. Define the events:

\begin{aligned} & A = \left\{ \text{ all the children are of the same sex }\right\} \\ & B = \left\{ \text{ there is at most one boy }\right\} \\ & C = \left\{ \text{ the family includes a boy and a girl }\right\} \end{aligned}

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(a)

Show that $A$ is independent of $B$ , and that $B$ is independent of $C$ .

(b)

Is $A$ independent of $C$ ?

(c)

Do these results hold if boys and girls are not equally likely?

(d)

Do these results hold if Jane has four children?

Problem 1.5.8

You flip three fair coins. At least two are alike, and it is an evens chance that the third is a head or a tail. Therefore $\mathbb {P}\left(\text{ all alike }\right) = \frac{1}{2}$ . Do you agree?

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Problem 1.5.9

Two fair dice are rolled. Show that the event that their sum is 7 is independent of the score shown by the first die.

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Problem 1.5.10

Let $X$ and $Y$ be the scores on two fair dice taking values in the set $\left\{ 1,2, \ldots , 6\right\}$ . Let $A_{1} = \left\{ X+Y = 9\right\} , A_{2} = \left\{ X \in \left\{ 1,2,3\right\} \right\}$ , and $A_{3} = \left\{ X \in \left\{ 3,4,5\right\} \right\}$ . Show that

\mathbb {P}\left(A_{1} \cap A_{2} \cap A_{3}\right) = \mathbb {P}\left(A_{1}\right) \mathbb {P}\left(A_{2}\right) \mathbb {P}\left(A_{3}\right)

Are these three events independent?

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