2.04c Calculate binomial probabilities

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 .

7 A fair die has one face numbered 1, one face numbered 3, two faces numbered 5 and two faces numbered 6 .

1. Find the probability of obtaining at least 7 odd numbers in 8 throws of the die. The die is thrown twice. Let \(X\) be the sum of the two scores. The following table shows the possible values of \(X\). \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Second throw}

   		1	3	5	5	6	6
   \cline { 2 - 8 }	1	2	4	6	6	7	7
   	3	4	6	8	8	9	9
   First	5	6	8	10	10	11	11
   throw	5	6	8	10	10	11	11
   	6	7	9	11	11	12	12
   	6	7	9	11	11	12	12

   \end{table}
2. Draw up a table showing the probability distribution of \(X\).
3. Calculate \(\mathrm { E } ( X )\).
4. Find the probability that \(X\) is greater than \(\mathrm { E } ( X )\).

1 The mean number of defective batteries in packs of 20 is 1.6 . Use a binomial distribution to calculate the probability that a randomly chosen pack of 20 will have more than 2 defective batteries.

7 The weights, \(X\) grams, of bars of soap are normally distributed with mean 125 grams and standard deviation 4.2 grams.

1. Find the probability that a randomly chosen bar of soap weighs more than 128 grams.
2. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 128 ) = 0.7465\).
3. Five bars of soap are chosen at random. Find the probability that more than two of the bars each weigh more than 128 grams.

3 The times taken by students to get up in the morning can be modelled by a normal distribution with mean 26.4 minutes and standard deviation 3.7 minutes.

1. For a random sample of 350 students, find the number who would be expected to take longer than 20 minutes to get up in the morning.
2. 'Very slow' students are students whose time to get up is more than 1.645 standard deviations above the mean. Find the probability that fewer than 3 students from a random sample of 8 students are 'very slow'.

6

1. State three conditions that must be satisfied for a situation to be modelled by a binomial distribution. On any day, there is a probability of 0.3 that Julie's train is late.
2. Nine days are chosen at random. Find the probability that Julie's train is late on more than 7 days or fewer than 2 days.
3. 90 days are chosen at random. Find the probability that Julie's train is late on more than 35 days or fewer than 27 days.

5 On trains in the morning rush hour, each person is either a student with probability 0.36 , or an office worker with probability 0.22 , or a shop assistant with probability 0.29 or none of these.

1. 8 people on a morning rush hour train are chosen at random. Find the probability that between 4 and 6 inclusive are office workers.
2. 300 people on a morning rush hour train are chosen at random. Find the probability that between 31 and 49 inclusive are neither students nor office workers nor shop assistants.

2 A factory produces flower pots. The base diameters have a normal distribution with mean 14 cm and standard deviation 0.52 cm . Find the probability that the base diameters of exactly 8 out of 10 randomly chosen flower pots are between 13.6 cm and 14.8 cm .

3 In a large consignment of mangoes, 15\% of mangoes are classified as small, 70\% as medium and \(15 \%\) as large.

1. Yue-chen picks 14 mangoes at random. Find the probability that fewer than 12 of them are medium or large.
2. Yue-chen picks \(n\) mangoes at random. The probability that none of these \(n\) mangoes is small is at least 0.1 . Find the largest possible value of \(n\).

5 Screws are sold in packets of 15. Faulty screws occur randomly. A large number of packets are tested for faulty screws and the mean number of faulty screws per packet is found to be 1.2 .

1. Show that the variance of the number of faulty screws in a packet is 1.104 .
2. Find the probability that a packet contains at most 2 faulty screws. Damien buys 8 packets of screws at random.
3. Find the probability that there are exactly 7 packets in which there is at least 1 faulty screw.

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.

7 The faces of a biased die are numbered \(1,2,3,4,5\) and 6 . The probabilities of throwing odd numbers are all the same. The probabilities of throwing even numbers are all the same. The probability of throwing an odd number is twice the probability of throwing an even number.

1. Find the probability of throwing a 3 .
2. The die is thrown three times. Find the probability of throwing two 5 s and one 4 .
3. The die is thrown 100 times. Use an approximation to find the probability that an even number is thrown at most 37 times.

7

1. A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.
   1. Find on how many days of the year ( 365 days) the daily sales can be expected to exceed 3900 litres. The daily sales at another petrol station are \(X\) litres, where \(X\) is normally distributed with mean \(m\) and standard deviation 560. It is given that \(\mathrm { P } ( X > 8000 ) = 0.122\).
   2. Find the value of \(m\).
   3. Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6 randomly chosen days.
2. The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\sigma = \frac { 2 } { 3 } \mu\), find the probability that a random value of \(Y\) is less than \(2 \mu\).

7 A factory makes water pistols, \(8 \%\) of which do not work properly.

1. A random sample of 19 water pistols is taken. Find the probability that at most 2 do not work properly.
2. In a random sample of \(n\) water pistols, the probability that at least one does not work properly is greater than 0.9 . Find the smallest possible value of \(n\).
3. A random sample of 1800 water pistols is taken. Use an approximation to find the probability that there are at least 152 that do not work properly.
4. Justify the use of your approximation in part (iii).

3 Visitors to a Wildlife Park in Africa have independent probabilities of 0.9 of seeing giraffes, 0.95 of seeing elephants, 0.85 of seeing zebras and 0.1 of seeing lions.

1. Find the probability that a visitor to the Wildlife Park sees all these animals.
2. Find the probability that, out of 12 randomly chosen visitors, fewer than 3 see lions.
3. 50 people independently visit the Wildlife Park. Find the mean and variance of the number of these people who see zebras.

3 On any day at noon, the probabilities that Kersley is asleep or studying are 0.2 and 0.6 respectively.

1. Find the probability that, in any 7-day period, Kersley is either asleep or studying at noon on at least 6 days.
2. Use an approximation to find the probability that, in any period of 100 days, Kersley is asleep at noon on at most 30 days.

2 A fair triangular spinner has three sides numbered 1, 2, 3. When the spinner is spun, the score is the number of the side on which it lands. The spinner is spun four times.

1. Find the probability that at least two of the scores are 3 .
2. Find the probability that the sum of the four scores is 5 .

7 Each day Annabel eats rice, potato or pasta. Independently of each other, the probability that she eats rice is 0.75 , the probability that she eats potato is 0.15 and the probability that she eats pasta is 0.1 .

1. Find the probability that, in any week of 7 days, Annabel eats pasta on exactly 2 days.
2. Find the probability that, in a period of 5 days, Annabel eats rice on 2 days, potato on 1 day and pasta on 2 days.
3. Find the probability that Annabel eats potato on more than 44 days in a year of 365 days.

3 An experiment consists of throwing a biased die 30 times and noting the number of 4 s obtained. This experiment was repeated many times and the average number of 4 s obtained in 30 throws was found to be 6.21.

1. Estimate the probability of throwing a 4.
   ..................................................................................................................................... .
   \section*{Hence}
2. find the variance of the number of 4 s obtained in 30 throws,
3. find the probability that in 15 throws the number of 4 s obtained is 2 or more.

4 A fair tetrahedral die has faces numbered \(1,2,3,4\). A coin is biased so that the probability of showing a head when thrown is \(\frac { 1 } { 3 }\). The die is thrown once and the number \(n\) that it lands on is noted. The biased coin is then thrown \(n\) times. So, for example, if the die lands on 3 , the coin is thrown 3 times.

1. Find the probability that the die lands on 4 and the number of times the coin shows heads is 2 .
2. Find the probability that the die lands on 3 and the number of times the coin shows heads is 3 .
3. Find the probability that the number the die lands on is the same as the number of times the coin shows heads.

5 Blank CDs are packed in boxes of 30 . The probability that a blank CD is faulty is 0.04 . A box is rejected if more than 2 of the blank CDs are faulty.

1. Find the probability that a box is rejected.
2. 280 boxes are chosen randomly. Use an approximation to find the probability that at least 30 of these boxes are rejected.

7 Josie aims to catch a bus which departs at a fixed time every day. Josie arrives at the bus stop \(T\) minutes before the bus departs, where \(T \sim \mathrm {~N} \left( 5.3,2.1 ^ { 2 } \right)\).

1. Find the probability that Josie has to wait longer than 6 minutes at the bus stop.
   On \(5 \%\) of days Josie has to wait longer than \(x\) minutes at the bus stop.
2. Find the value of \(x\).
3. Find the probability that Josie waits longer than \(x\) minutes on fewer than 3 days in 10 days.
4. Find the probability that Josie misses the bus.

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).

3 Jake attempts the crossword puzzle in his daily newspaper every day. The probability that he will complete the puzzle on any given day is 0.75 , independently of all other days.

1. Find the probability that he will complete the puzzle at least three times over a period of five days.
   Kenny also attempts the puzzle every day. The probability that he will complete the puzzle on a Monday is 0.8 . The probability that he will complete it on a Tuesday is 0.9 if he completed it on the previous day and 0.6 if he did not complete it on the previous day.
2. Find the probability that Kenny will complete the puzzle on at least one of the two days Monday and Tuesday in a randomly chosen week.