Fair games and further distributions Problem set A

You pay $1.50 to play a game. Three fair coins are tossed and you receive $1 for each head.

1. (a) Find the expected payout.
2. (b) Is the game fair? Give a reason.

A discrete random variable X has \(P(X=x)=\frac{x^2+1}{18}\) for x = 0, 1, 2, 3.

1. (a) Write the probability distribution of X as a table.
2. (b) Find E(X).

A fair four-sided die (faces 1 to 4) is rolled and a fair three-sided spinner (sectors 1 to 3) is spun. X is the sum of the two scores. The table shows every outcome.

1. (a) Write the probability distribution of X as a table.
2. (b) Find the expected total score.

A weighted coin lands heads (H) with probability 0.4. It is tossed three times. X is the number of heads. The tree diagram shows the outcomes.

1. (a) Find the probability of exactly two heads.
2. (b) Find the probability distribution of X.
3. (c) Find E(X).

Two cards are dealt from a standard pack of 52 without replacement. X is the number of aces dealt.

1. (a) Find the probability distribution of X.
2. (b) Find E(X).

In a board game, two fair four-sided dice (faces 1 to 4) are rolled. The score X is the positive difference between the two numbers. The table shows every outcome.

1. (a) Write the probability distribution of X as a table.
2. (b) Find the expected score E(X).

A school raffle sells 200 tickets at $2 each. There is one prize of $150 and three prizes of $20.

1. (a) Find the expected prize for one ticket.
2. (b) Find the expected gain for a ticket buyer. Is the raffle fair?
3. (c) Find the organiser's total profit if all the tickets are sold.

A player draws one card at random from a standard pack. If it is a spade the player wins $6. Otherwise the player pays $n. Find the value of n that makes the game fair.

The number of times, X, a printer jams in a day has \(P(X=x)=k(5-x)\) for x = 0, 1, 2, 3, 4, where k is a constant.

1. (a) Find the value of k.
2. (b) Find P(X ≤ 1).
3. (c) Find E(X).

The table shows the probability distribution of X. The expected value of X is 2.45.

1. (a) Write down two equations in a and b.
2. (b) Find the values of a and b.

A discrete random variable X has \(P(X=x)=cx^2\) for x = 1, 2, 3, 4, where c is a constant.

1. (a) Find the value of c.
2. (b) Find P(X ≥ 3 | X ≥ 2).
3. (c) Find E(X).