2.04c Calculate binomial probabilities

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.

2 In a certain large college, \(22 \%\) of students own a car.

1. 3 students from the college are chosen at random. Find the probability that all 3 students own a car.
2. 16 students from the college are chosen at random. Find the probability that the number of these students who own a car is at least 2 and at most 4 .

6 In Questa, 60\% of the adults travel to work by car.

1. A random sample of 12 adults from Questa is taken. Find the probability that the number who travel to work by car is less than 10 .
2. A random sample of 150 adults from Questa is taken. Use an approximation to find the probability that the number who travel to work by car is less than 81 .
3. Justify the use of your approximation in part (b).

5 Every day Richard takes a flight between Astan and Bejin. On any day, the probability that the flight arrives early is 0.15 , the probability that it arrives on time is 0.55 and the probability that it arrives late is 0.3 .

1. Find the probability that on each of 3 randomly chosen days, Richard's flight does not arrive late.
2. Find the probability that for 9 randomly chosen days, Richard's flight arrives early at least 3 times.
3. 60 days are chosen at random. Use an approximation to find the probability that Richard's flight arrives early at least 12 times.

7 In the region of Arka, the total number of households in the three villages Reeta, Shan and Teber is 800 . Each of the households was asked about the quality of their broadband service. Their responses are summarised in the following table.

1. 1. Find the probability that a randomly chosen household is in Shan and has poor broadband service.
   2. Find the probability that a randomly chosen household has good broadband service given that the household is in Shan.
      In the whole of Arka there are a large number of households. A survey showed that \(35 \%\) of households in Arka have no broadband service.
2. 1. 10 households in Arka are chosen at random. Find the probability that fewer than 3 of these households have no broadband service.
   2. 120 households in Arka are chosen at random. Use an approximation to find the probability that more than 32 of these households have no broadband service.
      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 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.

6 Eli has four fair 4 -sided dice with sides labelled \(1,2,3,4\). He throws all four dice at the same time. The random variable \(X\) denotes the number of 2s obtained.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 3 } { 64 }\).
2. Complete the following probability distribution table for \(X\).

   \(x\)	0	1	2	3	4
   \(\mathrm { P } ( X = x )\)	\(\frac { 81 } { 256 }\)			\(\frac { 3 } { 64 }\)	\(\frac { 1 } { 256 }\)

3. Find \(\mathrm { E } ( X )\).
   Eli throws the four dice at the same time on 96 occasions.
4. Use an approximation to find the probability that he obtains at least two 2 s on fewer than 20 of these occasions.

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.

5 The lengths of Western bluebirds are normally distributed with mean 16.5 cm and standard deviation 0.6 cm . A random sample of 150 of these birds is selected.

1. How many of these 150 birds would you expect to have length between 15.4 cm and 16.8 cm ?
   The lengths of Eastern bluebirds are normally distributed with mean 18.4 cm and standard deviation \(\sigma \mathrm { cm }\). It is known that \(72 \%\) of Eastern bluebirds have length greater than 17.1 cm .
2. Find the value of \(\sigma\).
   A random sample of 120 Eastern bluebirds is chosen.
3. Use an approximation to find the probability that fewer than 80 of these 120 bluebirds have length greater than 17.1 cm .

2 Anil is a candidate in an election. He received \(40 \%\) of the votes. A random sample of 120 voters is chosen. Use an approximation to find the probability that, of the 120 voters, between 36 and 54 inclusive voted for Anil.

6 The mass of grapes sold per day by a large shop can be modelled by a normal distribution with mean 28 kg . On \(10 \%\) of days less than 16 kg of grapes are sold.

1. Find the standard deviation of the mass of grapes sold per day.
   The mass of grapes sold on any day is independent of the mass sold on any other day.
2. 12 days are chosen at random. Find the probability that less than 16 kg of grapes are sold on more than 2 of these 12 days.
3. In a random sample of 365 days, on how many days would you expect the mass of grapes sold to be within 1.3 standard deviations of the mean?

4 A game for two players is played using a fair 4-sided dice with sides numbered 1, 2, 3 and 4. One turn consists of throwing the dice repeatedly up to a maximum of three times. When a 4 is obtained, no further throws are made during that turn. A player who obtains a 4 in their turn scores 1 point.

1. Show that the probability that a player obtains a 4 in one turn is \(\frac { 37 } { 64 }\).
   Xeno and Yao play this game.
2. Find the probability that neither Xeno nor Yao score any points in their first two turns.
3. Xeno and Yao each have three turns. Find the probability that Xeno scores 2 more points than Yao. \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1548_376_349} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_67_1566_466_328} ........................................................................................................................................ ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_72_1570_735_324} \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_72_1570_826_324} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_77_1570_913_324} ........................................................................................................................................ . ......................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1570_1187_324} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_67_1570_1279_324} ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_62_1570_1462_324} .......................................................................................................................................... ......................................................................................................................................... . .......................................................................................................................................... .......................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_71_1570_1905_324} \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_74_1570_1994_324} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_76_1570_2083_324} ........................................................................................................................................ ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_74_1570_2359_324} ......................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1570_2542_324} \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1570_2631_324}

5 In a certain area in the Arctic the probability that it snows on any given day is 0.7 , independent of all other days.

1. Find the probability that in a week (7 days) it snows on at least five days.
   A week in which it snows on at least five days out of seven is called a 'white' week.
2. Find the probability that in three randomly chosen weeks at least one is a white week.
   In a different area in the Arctic, the probability that a week is a white week is 0.8 .
3. Use a suitable approximation to find the probability that in 60 randomly chosen weeks fewer than 47 are white weeks.

6 The residents of Mahjing were asked to classify their local bus service:

• \(25 \%\) of residents classified their service as good.
• \(60 \%\) of residents classified their service as satisfactory.
• \(15 \%\) of residents classified their service as poor.
  1. A random sample of 110 residents of Mahjing is chosen.

Use a suitable approximation to find the probability that fewer than 22 residents classified their bus service as good.

For a random sample of 10 residents of Mahjing, find the probability that fewer than 8 classified their bus service as good or satisfactory.

Three residents of Mahjing are selected at random. Find the probability that one resident classified the bus service as good, one as satisfactory and one as poor.

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.

5 In Greenton, 70\% of the adults own a car. A random sample of 8 adults from Greenton is chosen.
[0pt]

1. Find the probability that the number of adults in this sample who own a car is less than 6 . [3]
   A random sample of 120 adults from Greenton is now chosen.
2. Use an approximation to find the probability that more than 75 of them own a car.

7 There are 400 students at a school in a certain country. Each student was asked whether they preferred swimming, cycling or running and the results are given in the following table. A student is chosen at random.

1. 1. Find the probability that the student prefers swimming.
   2. Determine whether the events 'the student is male' and 'the student prefers swimming' are independent, justifying your answer.
      On average at all the schools in this country \(30 \%\) of the students do not like any sports.
2. 1. 10 of the students from this country are chosen at random. Find the probability that at least 3 of these students do not like any sports.
   2. 90 students from this country are now chosen at random. Use an approximation to find the probability that fewer than 32 of them do not like any sports.
      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 In a certain country, the probability of more than 10 cm of rain on any particular day is 0.18 , independently of the weather on any other day.

1. Find the probability that in any randomly chosen 7-day period, more than 2 days have more than 10 cm of rain.
2. For 3 randomly chosen 7-day periods, find the probability that exactly two of these periods have at least one day with more than 10 cm of rain.

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.

6 At a company's call centre, \(90 \%\) of callers are connected immediately to a representative.
A random sample of 12 callers is chosen.

1. Find the probability that fewer than 10 of these callers are connected immediately.
   A random sample of 80 callers is chosen.
2. Use an approximation to find the probability that more than 69 of these callers are connected immediately.
3. Justify the use of your approximation in part (b).

5 Company \(A\) produces bags of sugar. An inspector finds that on average \(10 \%\) of the bags are underweight. 10 of the bags are chosen at random.

1. Find the probability that fewer than 3 of these bags are underweight.
   The weights of the bags of sugar produced by company \(B\) are normally distributed with mean 1.04 kg and standard deviation 0.06 kg .
2. Find the probability that a randomly chosen bag produced by company \(B\) weighs more than 1.11 kg . \(81 \%\) of the bags of sugar produced by company \(B\) weigh less than \(w \mathrm {~kg}\).
3. Find the value of \(w\).

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.