5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)

A book reviewer estimates that the probability that he receives a delivery of books to review on any one weekday is \(0.1\). The first weekday in September on which he receives a delivery of books to review is the \(X\)th weekday of September.

1. State an assumption needed for \(X\) to be well modelled by a geometric distribution. [1]
2. Find \(P(X = 11)\). [2]
3. Find \(P(X \leq 8)\). [2]
4. Find \(\text{Var}(X)\). [2]
5. Give a reason why a geometric distribution might not be an appropriate model for the first weekday in a calendar year on which the reviewer receives a delivery of books to review. [1]

The random variable \(D\) has the distribution Geo\((p)\). It is given that Var\((D) = \frac{40}{9}\). Determine

1. Var\((3D + 5)\). [1]
2. E\((3D + 5)\). [6]
3. \(\text{P}(D > \text{E}(D))\). [3]

A bag contains 3 green counters, 3 blue counters and \(w\) white counters. Counters are selected at random, one at a time, with replacement, until a white counter is drawn. The total number of counters selected, including the white counter, is denoted by \(X\).

1. In the case when \(w = 2\),
   1. write down the distribution of \(X\), [1]
   2. find \(P(3 < X \leq 7)\). [2]
2. In the case when E\((X) = 2\), determine the value of \(w\). [2]
3. In the case when \(w = 2\) and \(X = 6\), find the probability that the first five counters drawn alternate in colour. [2]

\begin{enumerate}[label=(\alph*)] \item In a game show contestants are asked up to five questions in succession to qualify for the next round. An incorrect answer eliminates a contestant from the game show. Let \(X\) denote the number of questions correctly answered by a contestant. The probability distribution of \(X\) is given below.

\(x\)	0	1	2	3	4	5
\(\mathrm{P}(X = x)\)	0.30	0.25	0.20	0.16	0.06	0.03

1. Find the expected number of correctly answered questions and the variance of the distribution. [3]
2. Find the probability that a randomly selected contestant will correctly answer 3 or more questions. [1]
3. Each show had two contestants. Find the probability that both the contestants will correctly answer at least one question. [2]

\item In a promotion, a newspaper included a token in every copy of the newspaper. A proportion, 0.002, are winning tokens and occur randomly. A reader keeps buying copies of the newspaper until he buys one with a winning token and then stops. Let \(Y\) denote the number of copies bought.

1. Explain briefly why this situation may be modelled by a geometric distribution and write down a formula for \(\mathrm{P}(Y = y)\). [2]
2. Find the probability that the reader gets a winning token with the twentieth copy bought. [2]
3. Find the probability that the reader will not have to buy more than three copies in order to get a winning token. [2] \end{enumerate]