5.02h Geometric: mean 1/p and variance (1-p)/p^2

A spinner can land on red or blue. When the spinner is spun, there is a probability of \(\frac{1}{3}\) that it lands on blue. The spinner is spun repeatedly. The random variable \(B\) represents the number of the spin when the spinner first lands on blue.

1. Find
   1. P(\(B = 4\))
   2. P(\(B \leq 5\))
   [4]
2. Find E(\(B^2\)) [3]

Steve invites Tamara to play a game with this spinner. Tamara must choose a colour, either red or blue. Steve will spin the spinner repeatedly until the spinner first lands on the colour Tamara has chosen. The random variable \(X\) represents the number of the spin when this occurs. If Tamara chooses red, her score is \(e^X\) If Tamara chooses blue, her score is \(X^2\)

1. State, giving your reasons and showing any calculations you have made, which colour you would recommend that Tamara chooses. [5]

The discrete random variable \(X\) has a geometric distribution. It is given that \(\text{Var}(X) = 20\). Determine \(P(X \geq 7)\). [6]

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]