5.02f Geometric distribution: conditions

1 A website simulates the outcome of throwing four fair dice. Ten thousand people take part in a challenge using the website in which they have one attempt at getting four sixes in the four throws of the dice. The number of people who succeed in getting four sixes is denoted by the random variable \(X\).

1. Show that, for each person, the probability that the person gets four sixes is equal to \(\frac { 1 } { 1296 }\).
2. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
3. Use a Poisson distribution to calculate each of the following probabilities.
   Determine the probability that no more than 2 people succeed in getting four sixes at least once in their 20 attempts.

3 A tennis player is practising her serve. Each time she serves, she has a \(55 \%\) chance of being successful (getting the serve in the required area without hitting the net). You should assume that whether she is successful on any serve is independent of whether she is successful on any other serve.

1. Find the probability that the player is not successful in any of her first three serves.
2. Determine the probability that the player is successful at least 10 times in her first 20 serves.
3. Determine the probability that the player is successful for the first time on her fifth serve.
4. Determine the probability that the player is successful for the fifth time on her tenth serve. Another player is also practising his serve. Each time he serves, he has a probability \(p\) of being successful. You should assume that whether he is successful on any serve is independent of whether he is successful on any other serve. The probability that he is successful for the first time on his second serve is 0.2496 and the probability that he is successful on both of his first two serves is less than 0.25 .
5. Determine the value of \(p\).

1. There are 32 students in a class.

Each student rolls a fair die repeatedly, stopping when their total number of sixes is 4 Each student records the total number of times they rolled the die. Estimate the probability that the mean number of rolls for the class is less than 27.2

1. The probability of winning a prize when playing a single game of Pento is \(\frac { 1 } { 5 }\)

When more than one game is played the games are independent.
Sam plays 20 games.

1. Find the probability that Sam wins 4 or more prizes. Tessa plays a series of games.
2. Find the probability that Tessa wins her 4th prize on her 20th game. Rama invites Sam and Tessa to play some new games of Pento. They must pay Rama \(\pounds 1\) for each game they play but Rama will pay them \(\pounds 2\) for the first time they win a prize, \(\pounds 4\) for the second time and \(\pounds ( 2 w )\) when they win their \(w\) th prize ( \(w > 2\) ) Sam decides to play \(n\) games of Pento with Rama.
3. Show that Sam's expected profit is \(\pounds \frac { 1 } { 25 } \left( n ^ { 2 } - 16 n \right)\) Given that Sam chose \(n = 15\)
4. find the probability that Sam does not make a loss. Tessa agrees to play Pento with Rama. She will play games until she wins \(r\) prizes and then she will stop.
5. Find, in terms of \(r\), Tessa's expected profit.

1. The probability of Richard winning a prize in a game at the fair is 0.15

Richard plays a number of games.

1. Find the probability of Richard winning his second prize on his 8th game,
2. State two assumptions that have to be made, for the model used in part (a) to be valid. M ary plays the same game, but has a different probability of winning a prize. She plays until she has won r prizes. The random variable \(G\) represents the total number of games M ary plays.
3. Given that the mean and standard deviation of G are 18 and 6 respectively, determine whether Richard or Mary has the greater probability of winning a prize in a game.

1. Sam and Tessa are testing a spinner to see if the probability, p , of it landing on red is less than \(\frac { 1 } { 5 }\). They both use a \(10 \%\) significance level.

Sam decides to spin the spinner 20 times and record the number of times it lands on red.

1. Find the critical region for Sam's test.
2. Write down the size of Sam's test. Tessa decides to spin the spinner until it lands on red and she records the number of spins.
3. Find the critical region for Tessa's test.
4. Find the size of Tessa's test.
5. 1. Show that the power function for Sam's test is given by $$( 1 - p ) ^ { 19 } ( 1 + 19 p )$$
   2. Find the power function for Tessa's test.
6. With reference to parts (b), (d) and (e), state, giving your reasons, whether you would recommend Sam's test or Tessa's test when \(\mathrm { p } = 0.15\)

1 A learner driver keeps taking the driving test until she passes. The number of attempts taken, up to and including the pass, is denoted by \(X\).

1. State two assumptions needed for \(X\) to be well modelled by a geometric distribution. Assume now that \(X \sim \operatorname { Geo } ( 0.4 )\).
2. Find \(\mathrm { P } ( X < 6 )\).
3. Find \(\mathrm { E } ( X )\).
4. Find \(\operatorname { Var } ( X )\).

2 Shooting stars occur randomly, independently of one another and at a constant average rate of 12.0 per hour. On each of a series of randomly chosen clear nights I look for shooting stars for 20 minutes at a time. A successful night is a night on which I see at least 8 shooting stars in a 20 -minute period.
From tomorrow, I will count the number, \(X\), of nights on which I look for shooting stars, up to and including the first successful night. Find \(\mathrm { E } ( X )\).

4 A fair six-sided dice is rolled repeatedly. Find the probability of the following events.

1. A five occurs for the first time on the fourth roll.
2. A five occurs at least once in the first four rolls.
3. A five occurs for the second time on the third roll.
4. At least two fives occur in the first three rolls. The dice is rolled repeatedly until a five occurs for the second time.
5. Find the expected number of rolls required for two fives to occur. Justify your answer.

2. Mary and Jeff are archers and one morning they play the following game. They shoot an arrow at a target alternately, starting with Mary. The winner is the first to hit the target. You may assume that, with each shot, Mary has a probability 0.25 of hitting the target and Jeff has a probability \(p\) of hitting the target. Successive shots are independent.

1. Determine the probability that Jeff wins the game
   i) with his first shot,
   ii) with his second shot.
2. Show that the probability that Jeff wins the game is $$\frac { 3 p } { 1 + 3 p }$$
3. Find the range of values of \(p\) for which Mary is more likely to win the game than Jeff.

2 A bag contains four black balls and one white ball. A man chooses a ball at random. If it is a black ball, he replaces it and chooses another at random. If he chooses the white ball, he stops.

1. Name the probability distribution which models this situation.
2. Calculate the probability that he will make exactly three attempts before he stops.
3. Calculate the probability that he will make fewer than three attempts before he stops.

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.

The probability that a particular type of light bulb is defective is 0.01. A large number of these bulbs are tested, one by one. Assuming independence, find the probability that the tenth bulb tested is the first to be found defective. [2] The first defective bulb is the \(N\)th to be tested. Write down the value of E\((N)\). [1] Find the least value of \(n\) such that P\((N < n)\) is greater than 0.9. [3]

The probability that a particular type of light bulb is defective is \(0.01\). A large number of these bulbs are tested, one by one. Assuming independence, find the probability that the tenth bulb tested is the first to be found defective. [2] The first defective bulb is the \(N\)th to be tested. Write down the value of E(\(N\)). [1] Find the least value of \(n\) such that P(\(N \leqslant n\)) is greater than \(0.9\). [3]

A fair die is thrown repeatedly until a 6 is obtained.

1. Find the probability that obtaining a 6 takes no more than four throws. [2]
2. Find the least integer \(N\) such that the probability of obtaining a 6 before the \(N\)th throw is more than 0.95. [3]

A fair die is thrown repeatedly until a 6 is obtained.

1. Find the probability that obtaining a 6 takes no more than four throws. [2]
2. Find the least integer \(N\) such that the probability of obtaining a 6 before the \(N\)th throw is more than 0.95. [3]

A biased coin is tossed repeatedly until a head is obtained. The random variable \(X\) denotes the number of tosses required for a head to be obtained. The mean of \(X\) is equal to twice the variance of \(X\). Show that the probability that a head is obtained when the coin is tossed once is \(\frac{2}{3}\). [2] Find

1. P(\(X = 4\)), [1]
2. P(\(X > 4\)), [2]
3. the least integer \(N\) such that P(\(X \leq N\)) \(> 0.999\). [3]

Lan starts a new job on Monday. He will catch the bus to work every day from Monday to Friday inclusive. The probability that he will get a seat on the bus has the constant value \(p\). The random variable \(X\) denotes the number of days that Lan will catch the bus until he is able to get a seat. The probability that Lan will not get a seat on the Monday, Tuesday, Wednesday or Thursday of his first week is 0.4096.

1. Show that \(p = 0.2\). [2]
2. Find the probability that Lan first gets a seat on Monday of the second week in his new job. [2]
3. Find the least integer \(N\) such that \(\text{P}(X \leqslant N) > 0.9\), and identify the day and the week that corresponds to this value of \(N\). [4]

80\% of the residents of Kinwawa are in favour of a leisure centre being built in the town. 20 residents of Kinwawa are chosen at random and asked, in turn, whether they are in favour of the leisure centre.

1. Find the probability that more than 17 of these residents are in favour of the leisure centre. [3]
2. Find the probability that the 5th person asked is the first person who is not in favour of the leisure centre. [1]
3. Find the probability that the 7th person asked is the second person who is not in favour of the leisure centre. [2]

1. Write down the conditions under which the binomial distribution may be a suitable model to use in statistical work. [4]

A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly. Find the probability that

1. the first 5 will occur on the sixth throw, [8]
2. in the first eight throws there will be exactly three 5s.