5.02f Geometric distribution: conditions

7 On any given day, the probability that Moena messages her friend Pasha is 0.72 .

1. Find the probability that for a random sample of 12 days Moena messages Pasha on no more than 9 days.
2. Moena messages Pasha on 1 January. Find the probability that the next day on which she messages Pasha is 5 January.
3. Use an approximation to find the probability that in any period of 100 days Moena messages Pasha on fewer than 64 days.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The random variable \(X\) denotes the number of throws required to obtain a pair of tails.

1. Find the expected value of \(X\).
2. Find the probability that exactly 3 throws are required to obtain a pair of tails.
3. Find the probability that fewer than 6 throws are required to obtain a pair of tails.
   On a different occasion, a pair of fair coins is thrown 80 times.
4. Use an approximation to find the probability that a pair of tails is obtained more than 25 times.

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.

4 Three fair six-sided dice, each with faces marked \(1,2,3,4,5,6\), are thrown at the same time, repeatedly. For a single throw of the three dice, the score is the sum of the numbers on the top faces.

1. Find the probability that the score is 4 on a single throw of the three dice.
2. Find the probability that a score of 18 is obtained for the first time on the 5th throw of the three dice.

7 A children's wildlife magazine is published every Monday. For the next 12 weeks it will include a model animal as a free gift. There are five different models: tiger, leopard, rhinoceros, elephant and buffalo, each with the same probability of being included in the magazine. Sahim buys one copy of the magazine every Monday.

1. Find the probability that the first time that the free gift is an elephant is before the 6th Monday.
2. Find the probability that Sahim will get more than two leopards in the 12 magazines.
3. Find the probability that after 5 weeks Sahim has exactly one of each animal.
   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 A fair 5 -sided spinner has sides labelled 1, 2, 3, 4, 5. The spinner is spun repeatedly until a 2 is obtained on the side on which the spinner lands. The random variable \(X\) denotes the number of spins required.

1. Find \(\mathrm { P } ( X = 4 )\).
2. Find \(\mathrm { P } ( X < 6 )\).
   Two fair 5 -sided spinners, each with sides labelled \(1,2,3,4,5\), are spun at the same time. If the numbers obtained are equal, the score is 0 . Otherwise, the score is the higher number minus the lower number.
3. Find the probability that the score is greater than 0 given that the score is not equal to 2 .
   The two spinners are spun at the same time repeatedly .
4. For 9 randomly chosen spins of the two spinners, find the probability that the score is greater than 2 on at least 3 occasions.

1 Two fair coins are thrown at the same time repeatedly until a pair of heads is obtained. The number of throws taken is denoted by the random variable \(X\).

1. State the value of \(\mathrm { E } ( X )\).
2. Find the probability that exactly 5 throws are required to obtain a pair of heads.
3. Find the probability that fewer than 7 throws are required to obtain a pair of heads.

1 Rajesh applies once every year for a ticket to a music festival. The probability that he is successful in any particular year is 0.3 , independently of other years.

1. Find the probability that Rajesh is successful for the first time on his 7th attempt.
2. Find the probability that Rajesh is successful for the first time before his 6th attempt.
3. Find the probability that Rajesh is successful for the second time on his 10th attempt.

5 Salah decides to attempt the crossword puzzle in his newspaper each day. The probability that he will complete the puzzle on any given day is 0.65 , independent of other days.
[0pt]

1. Find the probability that Salah completes the puzzle for the first time on the 5th day. [1]
2. Find the probability that Salah completes the puzzle for the second time on the 5th day.
3. Find the probability that Salah completes the puzzle fewer than 5 times in a week (7 days). [3] \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-10_2713_31_145_2014}
4. Use a suitable approximation to find the probability that Salah completes the puzzle more than 50 times in a period of 84 days.

2 An ordinary fair die is thrown repeatedly until a 1 or a 6 is obtained.

1. Find the probability that it takes at least 3 throws but no more than 5 throws to obtain a 1 or a 6 .
   On another occasion, this die is thrown 3 times. The random variable \(X\) is the number of times that a 1 or a 6 is obtained.
2. Draw up the probability distribution table for \(X\).
3. Find \(\mathrm { E } ( X )\).

1 A fair spinner with 5 sides numbered 1,2,3,4,5 is spun repeatedly. The score on each spin is the number on the side on which the spinner lands.

1. Find the probability that a score of 3 is obtained for the first time on the 8th spin.
2. Find the probability that fewer than 6 spins are required to obtain a score of 3 for the first time.

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.

2 An ordinary fair die is thrown until a 6 is obtained.

1. Find the probability that obtaining a 6 takes more than 8 throws.
   Two ordinary fair dice are thrown together until a pair of 6s is obtained. The number of throws taken is denoted by the random variable \(X\).
2. Find the expected value of \(X\).
3. Find the probability that obtaining a pair of 6s takes either 10 or 11 throws.

1 Two fair coins are thrown at the same time. The random variable \(X\) is the number of throws of the two coins required to obtain two tails at the same time.

1. Find the probability that two tails are obtained for the first time on the 7th throw.
2. Find the probability that it takes more than 9 throws to obtain two tails for the first time.

6 In a game, Jim throws three darts at a board. This is called a 'turn'. The centre of the board is called the bull's-eye. The random variable \(X\) is the number of darts in a turn that hit the bull's-eye. The probability distribution of \(X\) is given in the following table. It is given that \(\mathrm { E } ( X ) = 0.55\).

1. Find the values of \(p\) and \(q\).
2. Find \(\operatorname { Var } ( X )\).
   Jim is practising for a competition and he repeatedly throws three darts at the board.
3. Find the probability that \(X = 1\) in at least 3 of 12 randomly chosen turns.
4. Find the probability that Jim first succeeds in hitting the bull's-eye with all three darts on his 9th turn.

5 The probability that a driver passes an advanced driving test is 0.3 on any given attempt.

1. Dipak keeps taking the test until he passes. The random variable \(X\) denotes the number of attempts required for Dipak to pass the test.
   1. Find \(\mathrm { P } ( 2 \leqslant X \leqslant 6 )\).
   2. Find \(\mathrm { E } ( X )\).
      Five friends will each take their advanced driving test tomorrow.
2. Find the probability that at least three of them will pass tomorrow.
   75 people will take their advanced driving test next week.
   [0pt]
3. Use an approximation to find the probability that more than 20 of them will pass next week. [5]

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}

7 In a game,players attempt to score a goal by kicking a ball into a net.The probability that Leno scores a goal is 0.4 on any attempt,independently of all other attempts.The random variable \(X\) denotes the number of attempts that it takes Leno to score a goal.

1. Find \(\mathrm { P } ( X = 5 )\) .
   ............................................................................................................................................
2. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\) .
3. Find the probability that Leno scores his second goal on or before his 5th attempt. \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-10_2715_33_106_2017} \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-11_2723_33_99_22} Leno has 75 attempts to score a goal.
4. Use a suitable approximation to find the probability that Leno scores more than 28 goals but fewer than 35 goals.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

5 A factory produces chocolates. 30\% of the chocolates are wrapped in gold foil, 25\% are wrapped in red foil and the remainder are unwrapped. Indigo chooses 8 chocolates at random from the production line.

1. Find the probability that she obtains no more than 2 chocolates that are wrapped in red foil.
   Jake chooses chocolates one at a time at random from the production line.
2. Find the probability that the first time he obtains a chocolate that is wrapped in red foil is before the 7th choice. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-08_2720_35_106_2015} \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-09_2717_29_105_22} Keifa chooses chocolates one at a time at random from the production line.
3. Find the probability that the second chocolate chosen is the first one wrapped in gold foil given that the fifth chocolate chosen is the first unwrapped chocolate.

1 The score when two fair six-sided dice are thrown is the sum of the two numbers on the upper faces.

1. Show that the probability that the score is 4 is \(\frac { 1 } { 12 }\).
   The two dice are thrown repeatedly until a score of 4 is obtained. The number of throws taken is denoted by the random variable \(X\).
2. Find the mean of \(X\).
3. Find the probability that a score of 4 is first obtained on the 6th throw.
4. Find \(\mathrm { P } ( X < 8 )\).

7 Passengers are travelling to Picton by minibus. The probability that each passenger carries a backpack is 0.65 , independently of other passengers. Each minibus has seats for 12 passengers.

1. Find the probability that, in a full minibus travelling to Picton, between 8 passengers and 10 passengers inclusive carry a backpack.
2. Passengers get on to an empty minibus. Find the probability that the fourth passenger who gets on to the minibus will be the first to be carrying a backpack.
3. Find the probability that, of a random sample of 250 full minibuses travelling to Picton, more than 54 will contain exactly 7 passengers carrying backpacks.

3 One plastic robot is given away free inside each packet of a certain brand of biscuits. There are four colours of plastic robot (red, yellow, blue and green) and each colour is equally likely to occur. Nick buys some packets of these biscuits. Find the probability that

1. he gets a green robot on opening his first packet,
2. he gets his first green robot on opening his fifth packet. Nick's friend Amos is also collecting robots.
3. Find the probability that the first four packets Amos opens all contain different coloured robots.

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.

4 A bag contains 6 white discs and 4 blue discs. Discs are removed at random, one at a time, without replacement.

1. Find the probability that
   1. the second disc is blue, given that the first disc was blue,
   2. the second disc is blue,
   3. the third disc is blue, given that the first disc was blue.
   4. The random variable \(X\) is the number of discs which are removed up to and including the first blue disc. State whether the variable X has a geometric distribution. Explain your answer briefly.

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