CAIE S1 (Statistics 1) 2012 November

1 Fabio drinks coffee each morning. He chooses Americano, Cappucino or Latte with probabilities 0.5, 0.3 and 0.2 respectively. If he chooses Americano he either drinks it immediately with probability 0.8 , or leaves it to drink later. If he chooses Cappucino he either drinks it immediately with probability 0.6 , or leaves it to drink later. If he chooses Latte he either drinks it immediately with probability 0.1 , or leaves it to drink later.

1. Find the probability that Fabio chooses Americano and leaves it to drink later.
2. Fabio drinks his coffee immediately. Find the probability that he chose Latte.

2 The random variable \(X\) is the daily profit, in thousands of dollars, made by a company. \(X\) is normally distributed with mean 6.4 and standard deviation 5.2.

1. Find the probability that, on a randomly chosen day, the company makes a profit between \(
   ) 10000\( and \)\\( 12000\).
2. Find the probability that the company makes a loss on exactly 1 of the next 4 consecutive days.

3 The table summarises the times that 112 people took to travel to work on a particular day.

1. State which time interval in the table contains the median and which time interval contains the upper quartile.
2. On graph paper, draw a histogram to represent the data.
3. Calculate an estimate of the mean time to travel to work.

4 The mean of a certain normally distributed variable is four times the standard deviation. The probability that a randomly chosen value is greater than 5 is 0.15 .

1. Find the mean and standard deviation.
2. 200 values of the variable are chosen at random. Find the probability that at least 160 of these values are less than 5 .

5

1. A team of 3 boys and 3 girls is to be chosen from a group of 12 boys and 9 girls to enter a competition. Tom and Henry are two of the boys in the group. Find the number of ways in which the team can be chosen if Tom and Henry are either both in the team or both not in the team.
2. The back row of a cinema has 12 seats, all of which are empty. A group of 8 people, including Mary and Frances, sit in this row. Find the number of different ways they can sit in these 12 seats if
   1. there are no restrictions,
   2. Mary and Frances do not sit in seats which are next to each other,
   3. all 8 people sit together with no empty seats between them.

6 A fair tetrahedral die has four triangular faces, numbered \(1,2,3\) and 4 . The score when this die is thrown is the number on the face that the die lands on. This die is thrown three times. The random variable \(X\) is the sum of the three scores.

1. Show that \(\mathrm { P } ( X = 9 ) = \frac { 10 } { 64 }\).
2. Copy and complete the probability distribution table for \(X\).

   \(x\)	3	4	5	6	7	8	9	10	11	12
   \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 64 }\)	\(\frac { 3 } { 64 }\)			\(\frac { 12 } { 64 }\)

3. Event \(R\) is 'the sum of the three scores is 9 '. Event \(S\) is 'the product of the three scores is 16 '. Determine whether events \(R\) and \(S\) are independent, showing your working.