Direct binomial probability calculation

7 James throws a discus repeatedly in an attempt to achieve a successful throw. A throw is counted as successful if the distance achieved is over 40 metres. For each throw, the probability that James is successful is \(\frac { 1 } { 4 }\), independently of all other throws. Find the probability that James takes

1. exactly 5 throws to achieve the first successful throw,
2. more than 8 throws to achieve the first successful throw. In order to qualify for a competition, a discus-thrower must throw over 40 metres within at most six attempts. When a successful throw is achieved, no further throws are taken. Find the probability that James qualifies for the competition. Colin is another discus-thrower. For each throw, the probability that he will achieve a throw over 40 metres is \(\frac { 1 } { 3 }\), independently of all other throws. Find the probability that exactly one of James and Colin qualifies for the competition.

1 In a certain country \(12 \%\) of houses have solar heating. 19 houses are chosen at random. Find the probability that fewer than 4 houses have solar heating.

6 On any occasion when a particular gymnast performs a certain routine, the probability that she will perform it correctly is 0.65 , independently of all other occasions.

1. Find the probability that she will perform the routine correctly on exactly 5 occasions out 7 .
2. On one day she performs the routine 50 times. Use a suitable approximation to estimate the probability that she will perform the routine correctly on fewer than 29 occasions.
3. On another day she performs the routine \(n\) times. Find the smallest value of \(n\) for which the expected number of correct performances is at least 8 .

3 The number of books read by members of a book club each year has the binomial distribution \(B ( 12,0.7 )\).

1. State the greatest number of books that could be read by a member of the book club in a particular year and find the probability that a member reads this number of books.
2. Find the probability that a member reads fewer than 10 books in a particular year.

1 In a certain town, 76\% of cars are fitted with satellite navigation equipment. A random sample of 11 cars from this town is chosen. Find the probability that fewer than 10 of these cars are fitted with this equipment.

5 At the Nonland Business College, all students sit an accountancy examination at the end of their first year of study. On average, \(80 \%\) of the students pass this examination.

1. A random sample of 9 students who will take this examination is chosen. Find the probability that at most 6 of these students will pass the examination.
2. A random sample of 200 students who will take this examination is chosen. Use a suitable approximate distribution to find the probability that more than 166 of them will pass the examination.
3. Justify the use of your approximate distribution in part (ii).

1 In a certain town, 76\% of cars are fitted with satellite navigation equipment. A random sample of 11 cars from this town is chosen. Find the probability that fewer than 10 of these cars are fitted with this equipment.

6 There are a large number of students in Luttley College. \(60 \%\) of the students are boys. Students can choose exactly one of Games, Drama or Music on Friday afternoons. It is found that \(75 \%\) of the boys choose Games, \(10 \%\) of the boys choose Drama and the remainder of the boys choose Music. Of the girls, \(30 \%\) choose Games, \(55 \%\) choose Drama and the remainder choose Music.

1. 6 boys are chosen at random. Find the probability that fewer than 3 of them choose Music.
2. 5 Drama students are chosen at random. Find the probability that at least 1 of them is a boy.
3. In a certain country, the daily minimum temperature, in \({ } ^ { \circ } \mathrm { C }\), in winter has the distribution \(\mathrm { N } ( 8,24 )\). Find the probability that a randomly chosen winter day in this country has a minimum temperature between \(7 ^ { \circ } \mathrm { C }\) and \(12 ^ { \circ } \mathrm { C }\). The daily minimum temperature, in \({ } ^ { \circ } \mathrm { C }\), in another country in winter has a normal distribution with mean \(\mu\) and standard deviation \(2 \mu\).
4. Find the proportion of winter days on which the minimum temperature is below zero.
5. 70 winter days are chosen at random. Find how many of these would be expected to have a minimum temperature which is more than three times the mean.
6. The probability of the minimum temperature being above \(6 ^ { \circ } \mathrm { C }\) on any winter day is 0.0735 . Find the value of \(\mu\).

6 Louise and Marie play a series of tennis matches. It is given that, in any match, the probability that Louise wins the first two sets is \(\frac { 3 } { 8 }\).

1. Find the probability that, in 5 randomly chosen matches, Louise wins the first two sets in exactly 2 of the matches. It is also given that Louise and Marie are equally likely to win the first set.
2. Show that P (Louise wins the second set, given that she won the first set) \(= \frac { 3 } { 4 }\).
3. The probability that Marie wins the first two sets is \(\frac { 1 } { 3 }\). Find P(Marie wins the second set, given that she won the first set).

4 When onion seeds are sown outdoors, on average two-thirds of them germinate. A gardener sows seeds in pairs, in the hope that at least one will germinate.

1. Assuming that germination of one of the seeds in a pair is independent of germination of the other seed, find the probability that, if a pair of seeds is selected at random,
   (A) both seeds germinate,
   (B) just one seed germinates,
   (C) neither seed germinates.
2. Explain why the assumption of independence is necessary in order to calculate the above probabilities. Comment on whether the assumption is likely to be valid.
3. A pair of seeds is sown. Find the expectation and variance of the number of seeds in the pair which germinate.
4. The gardener plants 200 pairs of seeds. If both seeds in a pair germinate, the gardener destroys one of the two plants so that only one is left to grow. Of the plants that remain after this, only \(85 \%\) successfully grow to form an onion. Find the expected number of onions grown from the 200 pairs of seeds. If the seeds are sown in a greenhouse, the germination rate is higher. The seed manufacturing company claims that the germination rate is \(90 \%\). The gardener suspects that the rate will not be as high as this, and carries out a trial to investigate. 18 randomly selected seeds are sown in the greenhouse and it is found that 14 germinate.
5. Write down suitable hypotheses and carry out a test at the \(5 \%\) level to determine whether there is any evidence to support the gardener's suspicions.

8 On average, half the plants of a particular variety produce red flowers and the rest produce blue flowers.

1. Ann chooses 8 plants of this variety at random. Find the probability that more than 6 plants produce red flowers.
2. Karim chooses 22 plants of this variety at random.
   1. Find the probability that the number of these plants that produce blue flowers is equal to the number that produce red flowers.
   2. Hence find the probability that the number of these plants that produce blue flowers is greater than the number that produce red flowers.

4 Each time Ben attempts to complete a crossword in his daily newspaper, the probability that he succeeds is \(\frac { 2 } { 3 }\). The random variable \(X\) denotes the number of times that Ben succeeds in 9 attempts.

1. Find
   1. \(\mathrm { P } ( X = 6 )\),
   2. \(\mathrm { P } ( X < 6 )\),
   3. \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\). Ben notes three values, \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), of \(X\).
   4. State the total number of attempts to complete a crossword that are needed to obtain three values of \(X\). Hence find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } = 18 \right)\).

2 A university has a large number of students, of whom \(35 \%\) are studying science subjects. A sample of 10 students is obtained by listing all the students, giving each a serial number and selecting by using random numbers.

1. Find the probability that fewer than 3 of the sample are studying science subjects.
2. It is required that, in selecting the sample, the same student is not selected twice. Explain whether this requirement invalidates your calculation in part (i).

1. A fair 5 -sided spinner has sides numbered \(1,2,3,4\) and 5

The spinner is spun once and the score of the side it lands on is recorded.

1. Write down the name of the distribution that can be used to model the score of the side it lands on. The spinner is spun 28 times.
   The random variable \(X\) represents the number of times the spinner lands on 2
2. 1. Find the probability that the spinner lands on 2 at least 7 times.
   2. Find \(\mathrm { P } ( 4 \leqslant X < 8 )\)

8 Every morning before breakfast Laura and Mike play a game of chess. The probability that Laura wins is 0.7 . The outcome of any particular game is independent of the outcome of other games. Calculate the probability that, in the next 20 games,

1. Laura wins exactly 14 games,
2. Laura wins at least 14 games.

5 It is known that 40\% of people in Britain carry a certain gene.
A random sample of 32 people is collected.

1. Calculate the probability that exactly 12 people carry the gene.
2. Calculate the probability that at least 8 people carry the gene, giving your answer correct to \(\mathbf { 3 }\) decimal places.

5

1. A team of 9 is chosen at random from a class consisting of 8 boys and 12 girls.
   Find the probability that the team contains no more than 3 girls.
2. A group of \(n\) people, including Mr and Mrs Laplace, are arranged at random in a line. The probability that Mr and Mrs Laplace are placed next to each other is less than 0.1 . Find the smallest possible value of \(n\).

1. A salesman sells insurance to people. Each day he chooses a number of people to contact. The probability that the salesman sells insurance to a person he contacts is 0.05

On Monday he chooses to contact 10 people.

1. Find the probability that on Monday the salesman sells insurance to
   1. exactly 1 person,
   2. at least 3 people.
2. Find the number of people he should contact each day in order to sell insurance, on average, to 3 people per day.
3. Calculate the least number of people he must choose to contact on Friday, so that the probability of selling insurance to at least 1 person on Friday exceeds 0.99

1. A cadet fires shots at a target at distances ranging from 25 m to 90 m . The probability of hitting the target with a single shot is \(p\). When firing from a distance \(d \mathrm {~m} , p = \frac { 3 } { 200 } ( 90 - d )\). Each shot is fired independently.

The cadet fires 10 shots from a distance of 40 m .

1. 1. Find the probability that exactly 6 shots hit the target.
   2. Find the probability that at least 8 shots hit the target. The cadet fires 20 shots from a distance of \(x \mathrm {~m}\).
2. Find, to the nearest integer, the value of \(x\) if the cadet has an \(80 \%\) chance of hitting the target at least once. The cadet fires 100 shots from 25 m .
3. Using a suitable approximation, estimate the probability that at least 95 of these shots hit the target.

7 A travel agency in Tunisia offers customers a 3-day tour into the Sahara desert by either coach or minibus.

1. The agency accepts bookings from 50 customers for seats on the coach. The probability that a customer, who has booked a seat on the coach, will not turn up to claim the seat is 0.08 , and may be assumed to be independent of the behaviour of other customers. Determine the probability that, of the customers who have booked a seat on the coach:
   1. two or more will not turn up;
   2. three or more will not turn up.
2. The agency accepts bookings from 15 customers for seats on the minibus. The probability that a customer, who has booked a seat on the minibus, will not turn up to claim the seat is 0.025 , and may be assumed to be independent of the behaviour of other customers. Calculate the probability that, of the customers who have booked a seat on the minibus:
   1. all will turn up;
   2. one or more will not turn up.
3. The coach has 48 seats and the minibus has 14 seats. If 14 or fewer customers who have booked seats on the minibus turn up, they will be allocated a seat on the minibus. If all 15 customers who have booked seats on the minibus turn up, one will be allocated a seat on the coach. This will leave only 47 seats available for the 50 customers who have booked seats on the coach. Use your results from parts (a) and (b) to calculate the probability that there will be seats available on the coach for all those who turn up having booked such seats.
   (4 marks)

6 A bin contains a very large number of paper clips of different colours. The proportion of each colour is shown in the table.

1. Packets are filled from the bin. Each filled packet contains exactly 30 paper clips which may be considered to be a random sample. Use binomial distributions to determine the probability that a filled packet contains:
   1. exactly 2 purple paper clips;
   2. a total of more than 10 red or purple paper clips;
   3. at least 5 but at most 10 green paper clips.
2. Jumbo packets are also filled from the bin. Each filled jumbo packet contains exactly 100 paper clips.
   1. Assuming that the number of paper clips in a jumbo packet may be considered to be a random sample, calculate the mean and the variance of the number of red paper clips in a filled jumbo packet.
   2. It is claimed that the proportion of red paper clips in the bin is greater than 0.22 and that jumbo packets do not contain random samples of paper clips. An analysis of the number of red paper clips in each of a random sample of 50 filled jumbo packets resulted in a mean of 22.1 and a standard deviation of 4.17. Comment, with numerical justification, on each of the two claims.

5 An analysis of the number of vehicles registered by each household within a city resulted in the following information.

1. A random sample of 30 households within the city is selected. Use a binomial distribution with \(n = 30\), together with relevant information from the table in each case, to find the probability that the sample contains:
   1. exactly 3 households with no registered vehicles;
   2. at most 5 households with three or more registered vehicles;
   3. more than 10 households with at least two registered vehicles;
   4. more than 5 households but fewer than 10 households with exactly two registered vehicles.
2. If a random sample of \(\mathbf { 1 5 0 }\) households within the city were to be selected, estimate the mean and the variance for the number of households in the sample that would have either one or two registered vehicles.
   [0pt] [2 marks]
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6 The proportions of different colours of loom bands in a box of 10000 loom bands are given in the table.

1. A sample of 50 loom bands is selected at random from the box. Use a binomial distribution with \(n = 50\), together with relevant information from the table, to estimate the probability that this sample contains:
   1. exactly 4 red loom bands;
   2. at most 10 yellow loom bands;
   3. at least 30 blue or green loom bands;
   4. more than 35 but fewer than 45 loom bands that are neither yellow nor white.
2. The random variable \(R\) denotes the number of red loom bands in a random sample of \(\mathbf { 3 0 0 }\) loom bands selected from the box. Estimate values for the mean and the variance of \(R\).
   [0pt] [2 marks]

3 A tennis player is practising her serve. Each time she serves, she has a \(55 \%\) chance of being successful (getting the serve in the required area without hitting the net). You should assume that whether she is successful on any serve is independent of whether she is successful on any other serve.

1. Find the probability that the player is not successful in any of her first three serves.
2. Determine the probability that the player is successful at least 10 times in her first 20 serves.
3. Determine the probability that the player is successful for the first time on her fifth serve.
4. Determine the probability that the player is successful for the fifth time on her tenth serve. Another player is also practising his serve. Each time he serves, he has a probability \(p\) of being successful. You should assume that whether he is successful on any serve is independent of whether he is successful on any other serve. The probability that he is successful for the first time on his second serve is 0.2496 and the probability that he is successful on both of his first two serves is less than 0.25 .
5. Determine the value of \(p\).

2 A hotel has 50 single rooms, 16 of which are on the ground floor. The hotel offers guests a choice of a full English breakfast, a continental breakfast or no breakfast. The probabilities of these choices being made are \(0.45,0.25\) and 0.30 respectively. It may be assumed that the choice of breakfast is independent from guest to guest.

1. On a particular morning there are 16 guests, each occupying a single room on the ground floor. Calculate the probability that exactly 5 of these guests require a full English breakfast.
2. On a particular morning when there are 50 guests, each occupying a single room, determine the probability that:
   1. at most 12 of these guests require a continental breakfast;
   2. more than 10 but fewer than 20 of these guests require no breakfast.
3. When there are 40 guests, each occupying a single room, calculate the mean and the standard deviation for the number of these guests requiring breakfast.