Sampling distribution of range or maximum

5. A bag contains a large number of coins. It contains only \(1 \mathrm { p } , 5 \mathrm { p }\) and 10 p coins. The fraction of 1 p coins in the bag is \(q\), the fraction of 5 p coins in the bag is \(r\) and the fraction of 10p coins in the bag is \(s\). Two coins are selected at random from the bag and the coin with the highest value is recorded. Let \(M\) represent the value of the highest coin. The sampling distribution of \(M\) is given below

1. List all the possible samples of two coins which may be selected.
2. Find the value of \(q\), the value of \(r\) and the value of \(s\)

6. At a men's tennis tournament there are 3 , 4 or 5 sets in a match. Over many years, data collected show that 50\% of matches last for exactly 3 sets, 30\% of matches last for exactly 4 sets and 20\% of matches last for exactly 5 sets. A random sample of 3 tennis matches is taken. The number of sets in each match is recorded as \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\) respectively. The random variable \(M\) represents the maximum value of \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\)

1. List all the samples where \(M \neq 5\)
2. Find the sampling distribution of \(M\)
3. Write down the mode of \(S _ { 1 }\) and the mode of \(M\)

1. A bag contains a large number of balls, each with one of the numbers 1,2 or 5 written on it in the ratio \(2 : 3 : 4\) respectively.

A random sample of 3 balls is taken from the bag.
The random variable \(B\) represents the range of the numbers written on the balls in the sample.

1. Find \(\mathrm { P } ( B = 4 )\)
2. Find the sampling distribution of \(B\).

6. A bag contains a large number of balls. 65\% are numbered 1 35\% are numbered 2 A random sample of 3 balls is taken from the bag.
Find the sampling distribution for the range of the numbers on the 3 selected balls.