Dean Chalmers and Julian Gilbey Solutions for Chapter: The Binomial and Geometric Distributions, Exercise 8: EXERCISE 7D

The random variable $Y$ follows a geometric distribution. Given that $P\left(Y=1\right)=0.2$, find $E\left(Y\right)$.

Given that $S~Geo\left(p\right)$ and that $E\left(S\right)=4\frac{1}{2}$, find $P\left(S=2\right).$

A biased $4$ -sided die is numbered $1,3,5$ and $7$. The probability of obtaining each score is proportional to that score. Find the probability that the first prime number is obtained on the third roll of the die.

Sylvie and Thierry are members of a choir. The probabilities that they can sing a perfect high $C$ note on each attempt are $\frac{4}{7}$ and $\frac{5}{8}$, respectively. Find the probability that both Sylvie and Thierry succeed in singing a high $C$ note on their second attempts.

A standard deck of $52$ playing cards has an equal number of hearts, spades, clubs and diamonds. A deck is shuffled and a card is randomly selected. Let $X$ be the number of cards selected, up to and including the first diamond. Find the probability that: $X$ is equal to $E\left(X\right)$

A study reports that a particular gene in $0.2\%$ of all people is defective. $X$ is the number of randomly selected people, up to and including the first person that has this defective gene. Given that $P\left(X\le b\right)>0.865$, find $E\left(X\right)$ and find the smallest possible value of $b$.

Anouar and Zane play a game in which they take turns at tossing a fair coin. The first person to toss heads is the winner. Anouar tosses the coin first, and the probability that he wins the game is $0.5^1+0.5^3+0.5^5+0.5^7+\dots$ Find the probability that Zane wins the game.

Anouar and Zane play a game in which they take turns at tossing a fair coin. The first person to toss heads is the winner. Anouar tosses the coin first, and the probability that he wins the game is $0.5^1+0.5^3+0.5^5+0.5^7+\dots$ Describe the sequence of results represented by the value $0.5^5$ in this series. Find the probability that Anouar wins the game.