2.04d Normal approximation to binomial

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 .

2 On a production line making toys, the probability of any toy being faulty is 0.08 . A random sample of 200 toys is checked. Use a suitable approximation to find the probability that there are at least 15 faulty toys.

6 A box contains 4 pears and 7 oranges. Three fruits are taken out at random and eaten. Find the probability that

1. 2 pears and 1 orange are eaten, in any order,
2. the third fruit eaten is an orange,
3. the first fruit eaten was a pear, given that the third fruit eaten is an orange. There are 121 similar boxes in a warehouse. One fruit is taken at random from each box.
4. Using a suitable approximation, find the probability that fewer than 39 are pears.

2 On average, 2 apples out of 15 are classified as being underweight. Find the probability that in a random sample of 200 apples, the number of apples which are underweight is more than 21 and less than 35.

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.

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 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 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.

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.

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.

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).

6 The lifetimes, in hours, of a particular type of light bulb are normally distributed with mean 2000 hours and standard deviation \(\sigma\) hours. The probability that a randomly chosen light bulb of this type has a lifetime of more than 1800 hours is 0.96 .

1. Find the value of \(\sigma\).
   New technology has resulted in a new type of light bulb. It is found that on average one in five of these new light bulbs has a lifetime of more than 2500 hours.
2. For a random selection of 300 of these new light bulbs, use a suitable approximate distribution to find the probability that fewer than 70 have a lifetime of more than 2500 hours.
3. Justify the use of your approximate distribution in part (ii).

4 In Quarendon, \(66 \%\) of households are satisfied with the speed of their wifi connection.

1. Find the probability that, out of 10 households chosen at random in Quarendon, at least 8 are satisfied with the speed of their wifi connection.
2. A random sample of 150 households in Quarendon is chosen. Use a suitable approximation to find the probability that more than 84 are satisfied with the speed of their wifi connection. [5]

7 A competition is taking place between two choirs, the Notes and the Classics. There is a large audience for the competition.

• \(30 \%\) of the audience are Notes supporters.
• \(45 \%\) of the audience are Classics supporters.
• The rest of the audience are not supporters of either of these choirs.
• No one in the audience supports both of these choirs.

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 . \includegraphics[max width=\textwidth, alt={}, center]{34ae4f06-d485-4138-82d8-902b70f08995-10_51_1563_495_331}
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.

6 Computer breakdowns occur randomly on average once every 48 hours of use.

1. Calculate the probability that there will be fewer than 4 breakdowns in 60 hours of use.
2. Find the probability that the number of breakdowns in one year (8760 hours) of use is more than 200.
3. Independently of the computer breaking down, the computer operator receives phone calls randomly on average twice in every 24 -hour period. Find the probability that the total number of phone calls and computer breakdowns in a 60-hour period is exactly 4 .

3 The number of customers who visit a particular shop between 9.00 am and 10.00 am has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 5.2. Following some new advertising, the manager wishes to test whether the value of \(\lambda\) has increased. He chooses a random sample of 20 days and finds that the total number of customers who visited the shop between 9.00 am and 10.00 am on those days is 125 . Use an approximating distribution to test at the \(2.5 \%\) significance level whether the value of \(\lambda\) has increased.

1 In a game, a ball is thrown and lands in one of 4 slots, labelled \(A , B , C\) and \(D\). Raju wishes to test whether the probability that the ball will land in slot \(A\) is \(\frac { 1 } { 4 }\).

1. State suitable null and alternative hypotheses for Raju's test.
   The ball is thrown 100 times and it lands in slot \(A 15\) times.
2. Use a suitable approximating distribution to carry out the test at the \(2 \%\) significance level.

7 Customers arrive at a particular shop at random times. It has been found that the mean number of customers who arrive during a 5 -minute interval is 2.1 .

1. Find the probability that exactly 4 customers arrive during a 10 -minute interval.
2. Find the probability that at least 4 customers arrive during a 20 -minute interval.
3. Use a suitable approximating distribution to find the probability that fewer than 40 customers arrive during a 2-hour interval.
   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 Most plants of a certain type have three leaves. However, it is known that, on average, 1 in 10000 of these plants have four leaves, and plants with four leaves are called 'lucky'. The number of lucky plants in a random sample of 25000 plants is denoted by \(X\).

1. State, with a justification, an approximating distribution for \(X\), giving the values of any parameters.
   Use your approximating distribution to answer parts (b) and (c).
2. Find \(\mathrm { P } ( X \leqslant 3 )\).
3. Given that \(\mathrm { P } ( X = k ) = 2 \mathrm { P } ( X = k + 1 )\), find \(k\).
   The number of lucky plants in a random sample of \(n\) plants, where \(n\) is large, is denoted by \(Y\).
4. Given that \(\mathrm { P } ( Y \geqslant 1 ) = 0.963\), correct to 3 significant figures, use a suitable approximating distribution to find the value of \(n\).

5 Cars arrive at a fuel station at random and at a constant average rate of 13.5 per hour.

1. Find the probability that more than 4 cars arrive during a 20-minute period.
2. Use an approximating distribution to find the probability that the number of cars that arrive during a 12-hour period is between 150 and 160 inclusive.
   Independently of cars, trucks arrive at the fuel station at random and at a constant average rate of 3.6 per 15-minute period.
3. Find the probability that the total number of cars and trucks arriving at the fuel station during a 10 -minute period is more than 3 and less than 7 .

3 It is known that \(1.8 \%\) of children in a certain country have not been vaccinated against measles. A random sample of 200 children in this country is chosen.

1. Use a suitable approximating distribution to find the probability that there are fewer than 3 children in the sample who have not been vaccinated against measles.
2. Justify your approximating distribution.

4 The number of cars arriving at a certain road junction on a weekday morning has a Poisson distribution with mean 4.6 per minute. Traffic lights are installed at the junction and council officer wishes to test at the \(2 \%\) significance level whether there are now fewer cars arriving. He notes the number of cars arriving during a randomly chosen 2 -minute period.

1. State suitable null and alternative hypotheses for the test.
2. Find the critical region for the test.
   The officer notes that, during a randomly chosen 2 -minute period on a weekday morning, exactly 5 cars arrive at the junction.
3. Carry out the test.
4. State, with a reason, whether it is possible that a Type I error has been made in carrying out the test in part (c).
   The number of cars arriving at another junction on a weekday morning also has a Poisson distribution with mean 4.6 per minute.
5. Use a suitable approximating distribution to find the probability that more than 300 cars will arrive at this junction in an hour.

5 The number of clients who arrive at an information desk has a Poisson distribution with mean 2.2 per 5-minute period.

1. Find the probability that, in a randomly chosen 15 -minute period, exactly 6 clients arrive at the desk.
2. If more than 4 clients arrive during a 5 -minute period, they cannot all be served. Find the probability that, during a randomly chosen 5 -minute period, not all the clients who arrive at the desk can be served.
3. Use a suitable approximating distribution to find the probability that, during a randomly chosen 1-hour period, fewer than 20 clients arrive at the desk.