Normal approximation to binomial

6 The probability that New Year's Day is on a Saturday in a randomly chosen year is \(\frac { 1 } { 7 }\).

1. 15 years are chosen randomly. Find the probability that at least 3 of these years have New Year's Day on a Saturday.
2. 56 years are chosen randomly. Use a suitable approximation to find the probability that more than 7 of these years have New Year's Day on a Saturday.

7 A die is biased so that the probability of throwing a 5 is 0.75 and the probabilities of throwing a 1,2 , 3 , 4 or 6 are all equal.

1. The die is thrown three times. Find the probability that the result is a 1 followed by a 5 followed by any even number.
2. Find the probability that, out of 10 throws of this die, at least 8 throws result in a 5 .
3. The die is thrown 90 times. Using an appropriate approximation, find the probability that a 5 is thrown more than 60 times.

7 In a certain country, \(60 \%\) of mobile phones sold are made by Company \(A , 35 \%\) are made by Company \(B\) and 5\% are made by other companies.

1. Find the probability that, out of a random sample of 13 people who buy a mobile phone, fewer than 11 choose a mobile phone made by Company \(A\).
2. Use a suitable approximation to find the probability that, out of a random sample of 130 people who buy a mobile phone, at least 50 choose a mobile phone made by Company \(B\).
3. A random sample of \(n\) mobile phones sold is chosen. The probability that at least one of these phones is made by Company \(B\) is more than 0.98 . Find the least possible value of \(n\).
   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 In a certain country the probability that a child owns a bicycle is 0.65 .

1. A random sample of 15 children from this country is chosen. Find the probability that more than 12 own a bicycle.
2. A random sample of 250 children from this country is chosen. Use a suitable approximation to find the probability that fewer than 179 own a bicycle.

5 On average, \(34 \%\) of the people who go to a particular theatre are men.

1. A random sample of 14 people who go to the theatre is chosen. Find the probability that at most 2 people are men.
2. Use an approximation to find the probability that, in a random sample of 600 people who go to the theatre, fewer than 190 are men.

In Marumbo, three quarters of the adults own a cell phone.

1. A random sample of 8 adults from Marumbo is taken. Find the probability that the number of adults who own a cell phone is between 4 and 6 inclusive. [3]
2. A random sample of 160 adults from Marumbo is taken. Use an approximation to find the probability that more than 114 of them own a cell phone. [5]
3. Justify the use of your approximation in part (ii). [1]

A company always sends letters by second class post unless they are marked first class. Over a long period of time it has been established that 20\% of letters to be posted are marked first class. In a random selection of 10 letters to be posted, find the probability that the number marked first class is

1. at least 3, [2]
2. fewer than 2. [2]

One Monday morning there are only 12 first class stamps. Given that there are 70 letters to be posted that day,

1. use a suitable approximation to find the probability that there are enough first class stamps, [7]
2. State an assumption about these 70 letters that is required in order to make the calculation in part (c) valid. [1]

A manufacturer produces large quantities of coloured mugs. It is known from previous records that 6\% of the production will be green. A random sample of 10 mugs was taken from the production line.

1. Define a suitable distribution to model the number of green mugs in this sample. [1]
2. Find the probability that there were exactly 3 green mugs in the sample. [3]

A random sample of 125 mugs was taken.

1. Find the probability that there were between 10 and 13 (inclusive) green mugs in this sample, using
   1. a Poisson approximation, [3]
   2. a Normal approximation. [6]

Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2 Find the probability that, in 9 games, Bhim loses

1. exactly 3 of the games, [3]
2. fewer than half of the games. [2]

Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 Bhim and Joe agree to play a further 60 games.

1. Calculate the mean and variance for the number of these 60 games that Bhim loses. [2]
2. Using a suitable approximation calculate the probability that Bhim loses more than 4 games. [3]

As part of a business studies project, 8 groups of students are each randomly allocated 10 different shares from a listing of over 300 share prices in a newspaper. Each group has to follow the changes in the price of their shares over a 3-month period. At the end of the 3 months, 35\% of all the shares in the listing have increased in price and the rest have decreased.

1. Find the probability that, for the 10 shares of one group,
   1. exactly 6 have gone up in price,
   2. more than 5 have gone down in price. [5 marks]
2. Using a suitable approximation, find the probability that of the 80 shares allocated in total to the groups, more than 35 will have decreased in value. [6 marks]

The sales staff at an insurance company make house calls to prospective clients. Past records show that 30% of the people visited will take out a new policy with the company. On a particular day, one salesperson visits 8 people. Find the probability that, of these,

1. exactly 2 take out new policies, [3 marks]
2. more than 4 take out new policies. [2 marks]

The company awards a bonus to any salesperson who sells more than 50 policies in a month.

1. Using a suitable approximation, find the probability that a salesperson gets a bonus in a month in which he visits 150 prospective clients. [5 marks]