Geometric Distribution

5 Malik is playing a game in which he has to throw a 6 on a fair six-sided die to start the game. Find the probability that

1. Malik throws a 6 for the first time on his third attempt,
2. Malik needs at most ten attempts to throw a 6.

2 A random variable \(T\) has the distribution \(\operatorname { Geo } \left( \frac { 1 } { 5 } \right)\). Find

1. \(\mathrm { P } ( T = 4 )\),
2. \(\mathrm { P } ( T > 4 )\),
3. \(\mathrm { E } ( T )\).

7 The discrete random variable \(X\) has a geometric distribution with mean 4.
Find

1. \(\mathrm { P } ( X = 5 )\),
2. \(\mathrm { P } ( X \geqslant 5 )\),
3. the least integer \(N\) such that \(\mathrm { P } ( X \leqslant N ) > 0.9995\).

2 The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.

1. Find the probability that all the men are next to each other.
2. Find the probability that no two men are next to one another.

7 The probability that a driver passes an advanced driving test has a fixed value \(p\) for each attempt. A driver keeps taking the test until he passes. The random variable \(X\) denotes the number of attempts required for the driver to pass. The variance of \(X\) is 3.75 .

1. Show that \(15 p ^ { 2 } + 4 p - 4 = 0\) and hence find the value of \(p\).
2. Find \(\mathrm { P } ( X = 5 )\).
3. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\).

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 \(\operatorname { Var } ( X )\).

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

3 Three fair 6-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown at the same time repeatedly. The score on each throw is the sum of the numbers on the uppermost faces.

1. Find the probability that a score of 17 or more is first obtained on the 6th throw.
2. Find the probability that a score of 17 or more is obtained in fewer than 8 throws.

6 A pair of coins is thrown repeatedly until a pair of heads is obtained. The number of throws taken is denoted by the random variable \(X\). State the expected value of \(X\). Find the probability that

1. exactly 4 throws are required to obtain a pair of heads,
2. fewer than 6 throws are required to obtain a pair of heads.

1 A fair six-sided die, with faces marked \(1,2,3,4,5,6\), is thrown repeatedly until a 4 is obtained.

1. Find the probability that obtaining a 4 requires fewer than 6 throws.
   On another occasion, the die is thrown 10 times.
2. Find the probability that a 4 is obtained at least 3 times.

5 The discrete random variable \(X\) has a geometric distribution. It is given that \(\operatorname { Var } ( X ) = 20\).
Determine \(\mathrm { P } ( X \geqslant 7 )\).

6 Malik is playing a game in which he has to throw a 6 on a fair six-sided die to start the game. Find the probability that

1. Malik throws a 6 for the first time on his third attempt,
2. Malik needs at most ten attempts to throw a 6.

4. The discrete random variable \(X\) has a geometric distribution. It is given that \(\operatorname { Var } ( X ) = 20\). Determine \(\mathrm { P } ( X \geqslant 7 )\).

1 Nicola throws an ordinary fair six-sided dice. The random variable \(X\) is the number of throws that she takes to obtain a 6.

1. Find \(\mathrm { P } ( X < 8 )\).
2. Find the probability that Nicola obtains a 6 for the second time on her 8th throw.
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6 Tom and Ben play a game repeatedly. The probability that Tom wins any game is 0.3 . Each game is won by either Tom or Ben. Tom and Ben stop playing when one of them (to be called the champion) has won two games.

1. Find the probability that Ben becomes the champion after playing exactly 2 games.
2. Find the probability that Ben becomes the champion.
3. Given that Tom becomes the champion, find the probability that he won the 2nd game.

2 The random variable \(X\) has the distribution \(\operatorname { Geo } ( 0.2 )\). Find

1. \(\mathrm { P } ( X = 3 )\),
2. \(\mathrm { P } ( 3 \leqslant X \leqslant 5 )\),
3. \(\mathrm { P } ( X > 4 )\). Two independent values of \(X\) are found.
4. Find the probability that the total of these two values is 3 .

4 Alex is practising bowling at a cricket wicket. Every time she bowls a ball, she has a \(30 \%\) chance of hitting the wicket.

1. Assuming that successive bowls are independent, determine the probability that Alex first hits the wicket on her third attempt.
2. Determine the probability that Alex hits the wicket for the fourth time on her tenth attempt.

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

3 In air traffic management, air traffic controllers send radio messages to pilots. On receiving a message, the pilot repeats it back to the controller to check that it has been understood correctly. At a particular site, on average \(4 \%\) of messages sent by controllers are not repeated back correctly and so have been misunderstood. You should assume that instances of messages being misunderstood occur randomly and independently.

1. Find the probability that exactly 2 messages are misunderstood in a sequence of 50 messages.
2. Find the probability that in a sequence of messages, the 10th message is the first one which is misunderstood.
3. Find the probability that in a sequence of 20 messages, there are no misunderstood messages.
4. Determine the expected number of messages required for 3 of them to be misunderstood.
5. Determine the probability that in a sequence of messages, the 3rd misunderstood message is the 60th message in the sequence.

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

7. A bag contains 4 red and 2 blue balls, all of the same size. A ball is selected at random and removed from the bag. This is repeated until a blue ball is pulled out of the bag. The random variable \(B\) is the number of balls that have been removed from the bag.

1. Show that \(\mathrm { P } ( B = 2 ) = \frac { 4 } { 15 }\).
2. Find the probability distribution of \(B\).
3. Find \(\mathrm { E } ( B )\). The bag and the same 6 balls are used in a game at a funfair. One ball is removed from the bag at a time and a contestant wins 50 pence if one of the first two balls picked out is blue.
4. What are the expected winnings from playing this game once? For \(\pounds 1\), a contestant gets to play the game three times.
5. What is the expected profit or loss from the three games?

6 The diagrams illustrate all or part of the probability distributions of the discrete random variables \(V , W , X , Y\) and \(Z\).
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1. One of these variables has the distribution \(\operatorname { Geo } \left( \frac { 1 } { 2 } \right)\). State, with a reason, which variable this is.
2. One of these variables has the distribution \(\mathrm { B } \left( 4 , \frac { 1 } { 2 } \right)\). State, with reasons, which variable this is.
   \(760 \%\) of the voters at a certain polling station are women. Voters enter the polling station one at a time. The number of voters who enter, up to and including the first woman, is denoted by \(X\).

6 A fair six-sided die is thrown until a 3 or a 4 is obtained. The number of throws taken is denoted by the random variable \(X\).

1. State the mean value of \(X\).
2. Find the probability that obtaining a 3 or a 4 takes exactly 6 throws.
3. Find the probability that obtaining a 3 or a 4 takes more than 4 throws.
4. Find the greatest integer \(n\) such that the probability of obtaining a 3 or a 4 in fewer than \(n\) throws is less than 0.95.