CAIE S1 (Statistics 1) 2014 November

The 50 members of a club include both the club president and the club treasurer. All 50 members want to go on a coach tour, but the coach only has room for 45 people. In how many ways can 45 members be chosen if both the club president and the club treasurer must be included? [3]

Find the number of different ways that 6 boys and 4 girls can stand in a line if

1. all 6 boys stand next to each other, [3]
2. no girl stands next to another girl. [3]

1. Four fair six-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown. Find the probability that the numbers shown on the four dice add up to 5. [3]
2. Four fair six-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown on 7 occasions. Find the probability that the numbers shown on the four dice add up to 5 on exactly 1 or 2 of the 7 occasions. [4]

Sharik attempts a multiple choice revision question on-line. There are 3 suggested answers, one of which is correct. When Sharik chooses an answer the computer indicates whether the answer is right or wrong. Sharik first chooses one of the three suggested answers at random. If this answer is wrong he has a second try, choosing an answer at random from the remaining 2. If this answer is also wrong Sharik then chooses the remaining answer, which must be correct.

1. Draw a fully labelled tree diagram to illustrate the various choices that Sharik can make until the computer indicates that he has answered the question correctly. [4]
2. The random variable \(X\) is the number of attempts that Sharik makes up to and including the one that the computer indicates is correct. Draw up the probability distribution table for \(X\) and find E\((X)\). [4]

1. The time, \(X\) hours, for which people sleep in one night has a normal distribution with mean 7.15 hours and standard deviation 0.88 hours.
   1. Find the probability that a randomly chosen person sleeps for less than 8 hours in a night. [2]
   2. Find the value of \(q\) such that P\((X < q) = 0.75\). [3]
2. The random variable \(Y\) has the distribution N\((\mu, \sigma^2)\), where \(2\sigma = 3\mu\) and \(\mu \neq 0\). Find P\((Y > 4\mu)\). [3]

On a certain day in spring, the heights of 200 daffodils are measured, correct to the nearest centimetre. The frequency distribution is given below.

Height (cm)	\(4 - 10\)	\(11 - 15\)	\(16 - 20\)	\(21 - 25\)	\(26 - 30\)
Frequency	22	32	78	40	28

1. Draw a cumulative frequency graph to illustrate the data. [4]
2. 28\% of these daffodils are of height \(h\) cm or more. Estimate \(h\). [2]
3. You are given that the estimate of the mean height of these daffodils, calculated from the table, is 18.39 cm. Calculate an estimate of the standard deviation of the heights of these daffodils. [3]

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]