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.