Dean Chalmers and Julian Gilbey Solutions for Chapter: The Binomial and Geometric Distributions, Exercise 6: EXERCISE 7C

Decide whether or not it would be appropriate to model the distribution of $X$ by a geometric distribution in the following situation. In those cases for which it is not appropriate, give a reason. $X$ is the number of times that a grain of rice is dropped from a height of $2$ metres onto a chessboard, up to and including the first time that it comes to rest on a white square.

Decide whether or not it would be appropriate to model the distribution of $X$ by a geometric distribution in the following situation. In those cases for which it is not appropriate, give a reason. $X$ is the number of races in which an athlete competes during a year, up to and including the first race that he wins.

The random variable $T$ has a geometric distribution and it is given that $\frac{P\left(T=2\right)}{P\left(T=5\right)}=15.625.$ Find $P\left(T=3\right).$

$X~Geo\left(p\right)$ and $P\left(X=2\right)=0.2464.$ Given that $p<0.5$, find $P\left(X>3\right).$

Given that $X~Geo\left(p\right)$ and that $P\left(X\le 4\right)=\frac{2385}{2401}$, find $P\left(1\le X<4\right).$

Two ordinary fair dice are rolled simultaneously. Find the probability of obtaining: the first double on the fourth roll.

Two ordinary fair dice are rolled simultaneously. Find the probability of obtaining: the first pair of numbers with a sum of more than $10$ before the $10$ th roll.

$X~Geo\left(0.24\right)$ and $Y~Geo\left(0.25\right)$ are two independent random variables. Find the probability that $X+Y=4.$