5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)

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 Jacob has four coins. One of the coins is biased such that when it is thrown the probability of obtaining a head is \(\frac { 7 } { 10 }\). The other three coins are fair. Jacob throws all four coins once. The number of heads that he obtains is denoted by the random variable \(X\). The probability distribution table for \(X\) is as follows.

1. Show that \(a = \frac { 1 } { 5 }\) and find the values of \(b\) and \(c\).
2. Find \(\mathrm { E } ( X )\).
   Jacob throws all four coins together 10 times.
3. Find the probability that he obtains exactly one head on fewer than 3 occasions.
4. Find the probability that Jacob obtains exactly one head for the first time on the 7th or 8th time that he throws the 4 coins.

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 Sam is a member of a soccer club. She is practising scoring goals. The probability that Sam will score a goal on any attempt is 0.7 , independently of all other attempts.

1. Sam makes 10 attempts at scoring goals. Find the probability that Sam will score goals on fewer than 8 of these attempts.
2. Find the probability that Sam's first successful attempt will be before her 5th attempt.
3. Wei is a member of the same soccer club. He is also practising scoring goals. The probability that Wei will score a goal on any attempt is 0.6 , independently of all other attempts. Wei is going to keep making attempts until he scores 3 goals.
   Find the probability that he scores his third goal on his 7th attempt.

3 Kayla is competing in a throwing event. A throw is counted as a success if the distance achieved is greater than 30 metres. The probability that Kayla will achieve a success on any throw is 0.25 .

1. Find the probability that Kayla takes more than 6 throws to achieve a success.
2. Find the probability that, for a random sample of 10 throws, Kayla achieves at least 3 successes.

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.

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.

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.

4 Three fair 4-sided spinners each have sides labelled 1,2,3,4. The spinners are spun at the same time and the number on the side on which each spinner lands is recorded. The random variable \(X\) denotes the highest number recorded.

1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 7 } { 64 }\).
2. Complete the probability distribution table for \(X\).

   \(x\)	1	2	3	4
   \(\mathrm { P } ( X = x )\)		\(\frac { 7 } { 64 }\)	\(\frac { 19 } { 64 }\)

   On another occasion, one of the fair 4 -sided spinners is spun repeatedly until a 3 is obtained. The random variable \(Y\) is the number of spins required to obtain a 3 .
3. Find \(\mathrm { P } ( Y = 6 )\).
4. Find \(\mathrm { P } ( Y > 4 )\).

2 Hazeem repeatedly throws two ordinary fair 6-sided dice at the same time. On each occasion, the score is the sum of the two numbers that she obtains.

1. Find the probability that it takes exactly 5 throws of the two dice for Hazeem to obtain a score of 8 or more.
2. Find the probability that it takes no more than 4 throws of the two dice for Hazeem to obtain a score of 8 or more.
3. For 8 randomly chosen throws of the two dice, find the probability that Hazeem obtains a score of 8 or more on fewer than 3 occasions.

2 George has a fair 5 -sided spinner with sides labelled 1,2,3,4,5. He spins the spinner and notes the number on the side on which the spinner lands.

1. Find the probability that it takes fewer than 7 spins for George to obtain a 5 .
   George spins the spinner 10 times.
2. Find the probability that he obtains a 5 more than 4 times but fewer than 8 times.

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.