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.

1. An unbiased die has faces numbered 1 to 6 inclusive. The die is rolled and the number that appears on the uppermost face is recorded.
   1. State the probability of not recording a 6 in one roll of the die.
   The die is thrown until a 6 is recorded.
2. Find the probability that a 6 occurs for the first time on the third roll of the die.
   (3)

7 Amir is trying to thread a needle. On each attempt the probability that he is successful is 0.3 , independently of any other attempt. The random variable \(X\) represents the number of attempts that he takes to thread the needle.

1. Find \(\mathrm { P } ( X = 5 )\).
2. During the course of a day, Amir has to thread 6 needles. Determine the probability that it takes him more than 3 attempts to be successful for at least 4 of the 6 needles.
3. Amaya is also trying to thread a needle. On each attempt the probability that she is successful is \(p\), independently of any other attempt. The probability that Amaya takes 2 attempts to thread a particular needle is \(\frac { 28 } { 121 }\). Determine the possible values of \(p\).

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

1. P\((X = 5)\), [3]
2. P\((X > 5)\), [2]
3. the least integer \(N\) such that P\((X \leqslant N) > 0.9995\). [2]

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]

6 The score when two fair dice are thrown is the sum of the two numbers on the upper faces. Two fair dice are thrown repeatedly until a score of 6 is obtained. The number of throws taken is denoted by the random variable \(X\). Find the mean of \(X\). Find the least integer \(N\) such that the probability of obtaining a score of 6 in fewer than \(N\) throws is more than 0.95 .

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. \(\mathrm { E } ( T )\).

1 An ordinary fair die is thrown repeatedly until a 5 is obtained. The number of throws taken is denoted by the random variable \(X\).

1. Write down the mean of \(X\).
2. Find the probability that a 5 is first obtained after the 3rd throw but before the 8th throw.
3. Find the probability that a 5 is first obtained in fewer than 10 throws.

9

1. A random variable \(X\) has the distribution \(\operatorname { Geo } \left( \frac { 1 } { 5 } \right)\). Find
   1. \(\mathrm { E } ( \mathrm { X } )\),
   2. \(\mathrm { P } ( \mathrm { X } = 4 )\),
   3. \(P ( X > 4 )\).
   4. A random variable \(Y\) has the distribution \(\operatorname { Geo } ( p )\), and \(q = 1 - p\).
      (a) Show that \(P ( Y\) is odd \() = p + q ^ { 2 } p + q ^ { 4 } p + \ldots\).
      (b) Use the formula for the sum to infinity of a geometric progression to show that $$P ( Y \text { is odd } ) = \frac { 1 } { 1 + q }$$ {}
      7

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.

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

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

8 Henry makes repeated attempts to light his gas fire. He makes the modelling assumption that the probability that the fire will light on any attempt is \(\frac { 1 } { 3 }\). Let \(X\) be the number of attempts at lighting the fire, up to and including the successful attempt.

1. Name the distribution of \(X\), stating a further modelling assumption needed. In the rest of this question, you should use the distribution named in part (i).
2. Calculate
   1. \(\mathrm { P } ( X = 4 )\),
   2. \(\mathrm { P } ( X < 4 )\).
   3. State the value of \(\mathrm { E } ( X )\).
   4. Henry has to light the fire once a day, starting on March 1st. Calculate the probability that the first day on which fewer than 4 attempts are needed to light the fire is March 3rd.

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 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.

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.

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 .

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. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-02_2717_35_109_2012}

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.

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.

6 A factory produces chocolates in three flavours: lemon, orange and strawberry in the ratio \(3 : 5 : 7\) respectively. Nell checks the chocolates on the production line by choosing chocolates randomly one at a time.

1. Find the probability that the first chocolate with lemon flavour that Nell chooses is the 7th chocolate that she checks.
2. Find the probability that the first chocolate with lemon flavour that Nell chooses is after she has checked at least 6 chocolates.
   'Surprise' boxes of chocolates each contain 15 chocolates: 3 are lemon, 5 are orange and 7 are strawberry. Petra has a box of Surprise chocolates. She chooses 3 chocolates at random from the box. She eats each chocolate before choosing the next one.
3. Find the probability that none of Petra's 3 chocolates has orange flavour.
4. Find the probability that each of Petra's 3 chocolates has a different flavour.
5. Find the probability that at least 2 of Petra's 3 chocolates have strawberry flavour given that none of them has orange flavour.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

\(R\) and \(S\) are independent random variables each having the distribution Geo\((p)\).

1. Find P\((R = 1\) and \(S = 1)\) in terms of \(p\). [1]
2. Show that P\((R = 3\) and \(S = 3) = p^2q^4\), where \(q = 1 - p\). [1]
3. Use the formula for the sum to infinity of a geometric series to show that $$\text{P}(R = S) = \frac{p}{2-p}.$$ [5]

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.

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.

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?