Geometric with multiple success milestones

5 Each year Jack enters a ballot for a concert ticket. The probability that Jack will win a ticket in any particular year is 0.27 .

1. Find the probability that the first time Jack wins a ticket is
   1. on his 8th attempt,
   2. after his 8th attempt.
   3. Write down an expression for the probability that Jack wins a ticket on exactly 2 of his first 8 attempts, and evaluate this expression.
   4. Find the probability that Jack wins his 3rd ticket on his 9th attempt and his 4th ticket on his 12th attempt.

2 Jill is collecting picture cards given away in packets of a particular brand of breakfast cereal. There are five different cards in the complete set. Each packet contains one card which is equally likely to be any of the five cards in the set.

1. Find the probability that Jill has a complete set of cards from the first five packets that she buys.
2. At some point Jill needs just one more card to complete the set. Let \(X\) be the random variable that represents the number of additional packets that Jill will need to buy in order to complete the set.
   1. Write down the distribution of \(X\).
   2. State the expected number of additional packets that Jill will need to buy.
   3. Find the probability that Jill will need to buy at least 3 additional packets in order to complete the set.

5 The random variable \(X\) has a geometric distribution: \(X \sim \operatorname { Geo } ( p )\).

1. Show that \(\mathrm { P } ( X > n ) = q ^ { n }\), where \(q = 1 - p\). You are given that \(\mathrm { P } ( X \geqslant 4 ) = 0.216\).
2. Use the result given in part (i) to find the value of \(p\) and \(\mathrm { P } ( X \leqslant 8 )\).
3. Write down \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

11 During the 30-day month of April, the probability that it will rain on any given day is 0.25 .

1. Find the probability that the first rainy day in April is the 7th of April, explaining any modelling assumptions you have made.
2. Given that it does not rain on the first 6 days of April, find the probability that it first rains on the 10th of April.
3. The probability that it first rains on the \(n\)th of April and next on the ( \(n + 3\) )th of April is 0.02 , correct to 1 significant figure. Determine \(n\).
4. Determine the expected number of dry days in April, given that it first rains on the 8th of April.

A game is played with a token on a board with a grid printed on it. The token starts at the point \((0, 0)\) and moves in steps. Each step is either 1 unit in the positive \(x\)-direction with probability 0.8, or 1 unit in the positive \(y\)-direction with probability 0.2. The token stops when it reaches a point with a \(y\)-coordinate of 1. It is given that the token stops at \((X, 1)\).

1. 1. Find the probability that \(X = 10\). [2]
   2. Find the probability that \(X < 10\). [3]
2. Find the expected number of steps taken by the token. [2]
3. Hence, write down the value of E\((X)\). [1]

It is known that 8% of the population of a large city use a particular web browser. A researcher wishes to interview some people from the city who use this browser. He selects people at random, one at a time.

1. Find the probability that the first person that he finds who uses this browser is
   1. the third person selected, [3]
   2. the second or third person selected. [2]
2. Find the probability that at least one of the first 20 people selected uses this browser. [3]

A darts player is trying to hit the bullseye on a dart board. On each throw the probability that she hits it is \(0.05\), independently of any other throw.

1. Find the probability that she hits the bullseye for the first time on her \(10\)th throw. [2]
2. Find the probability that she does not hit the bullseye in her first \(10\) throws. [1]
3. Write down the expected number of throws which it takes her to hit the bullseye for the first time. [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]