Basic E(X) and Var(X) calculation

1 A fair coin is tossed 5 times and the number of heads is recorded.

1. The random variable \(X\) is the number of heads. State the mean and variance of \(X\).
2. The number of heads is doubled and denoted by the random variable \(Y\). State the mean and variance of \(Y\).

1. A bag contains a large number of counters with an odd number or an even number written on each.

Odd and even numbered counters occur in the ratio \(4 : 1\)
In a game a player takes a random sample of 4 counters from the bag.
The player scores
5 points for each counter taken that has an even number written on it
2 points for each counter taken that has an odd number written on it
The random variable \(X\) represents the total score, in points, from the 4 counters.

1. Find the sampling distribution of \(X\) A random sample of \(n\) sets of 4 counters is taken. The random variable \(Y\) represents the number of these \(n\) sets that have a total score of exactly 14
2. Calculate the minimum value of \(n\) such that \(\mathrm { P } ( Y \geqslant 1 ) > 0.95\)

7. In an orchard, all the trees are either apple or pear trees. There are four times as many apple trees as pear trees. Find the probability that, in a random sample of 10 trees, there are

1. equal numbers of apple and pear trees,
2. more than 7 apple trees. In a sample of 60 trees in the orchard,
3. find the expected number of pear trees.
4. Calculate the standard deviation of the number of pear trees and compare this result with the standard deviation of the number of apple trees.
5. Find the probability that exactly 35 in the sample of 60 trees are pear trees.
6. Find an approximate value for the probability that more than 15 of the 60 trees are pear trees.

1 In a game at a charity fair, a spinner is spun 4 times.
On each spin the chance that the spinner lands on a score of 5 is 0.2 .
The random variable \(X\) represents the number of spins on which the spinner lands on a score of 5 .

1. Find \(\mathrm { P } ( X = 3 )\).
2. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   One game costs \(\pounds 1\) to play and, for each spin that lands on a score of 5 , the player receives 50 pence.
3. 1. Find the expected total amount of money gained by a player in one game.
   2. Find the standard deviation of the total amount of money gained by a player in one game.

4. Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick. Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard.
Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.

1. 1. Find the mean number of times he falls off in a day.
2. (ii) Find the variance of the number of times he falls off in a day.
3. 1. Find the probability that, on a particular day, he falls off exactly 10 times.
4. (ii) Find the probability that, on a particular day, he falls off 5 or more times.
5. Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
6. 1. Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days.
7. (ii) Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days.

17 An archer is training for the Olympics.
Each of the archer's training sessions consists of 30 attempts to hit the centre of a target. The archer consistently hits the centre of the target with \(79 \%\) of their attempts.
It can be assumed that the number of times the centre of the target is hit in any training session can be modelled by a binomial distribution. 17

1. Find the mean of the number of times that the archer hits the centre of the target during a training session. 17
2. Find the probability that the archer hits the centre of the target exactly 22 times during a particular training session. 17
3. Find the probability that the archer hits the centre of the target 18 times or less during a particular training session.
   [0pt] [1 mark] 17
4. Find the probability that the archer hits the centre of the target more than 26 times in a training session.
   [0pt] [2 marks]

5. Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick. Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard. Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.

1. 1. Find the mean number of times he falls off in a day.
2. (ii) Find the variance of the number of times he falls off in a day.
3. 1. Find the probability that, on a particular day, he falls off exactly 10 times.
4. (ii) Find the probability that, on a particular day, he falls off 5 or more times.
5. Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
6. 1. Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days.
7. (ii) Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days.