Exercises: p-values and expected value

A fair six-sided die is rolled once. Let $X$ be the score. Calculate the expected value $E(X)$.

A raffle in a Limerick GAA club sells 200 tickets at €2 each. There is one prize of €100 and two prizes of €25. Find the expected prize value of one ticket and the expected profit of a ticket holder.

A coin is tossed 10 times and gives 9 heads. Assuming the coin is fair, find the probability of obtaining 9 or more heads. Give your answer to 4 decimal places.

A game at a fundraiser in Sligo costs €3 to play. A player rolls two fair dice and wins €12 if the total is 7, €5 if the total is 11 or 12, and nothing otherwise.

1. a) Find the probability of each outcome.
2. b) Find the expected winnings and expected profit per game.
3. c) Should the organisers expect to make a profit? Explain.

A manufacturer in Waterford claims that 90% of its light bulbs last more than 1,000 hours. A sample of 20 bulbs is tested and 15 last more than 1,000 hours.

1. a) State suitable null and alternative hypotheses.
2. b) Find the p-value, $P(X\le 15)$, where $X\sim B(20, 0.9)$, to 4 decimal places.
3. c) State your conclusion at the 5% significance level.

The random variable $X$ has the probability distribution $P(X=x)=kx$ for $x = 1,2,3,4$.

1. a) Find the value of $k$.
2. b) Calculate $E(X)$.
3. c) Calculate $E(X^2)$ and the variance of $X$.

A school in Kilkenny claims that a student chosen at random has probability 0.3 of cycling to school. In a random sample of 50 students, 24 cycle.

1. a) State the hypotheses for a test of whether the true proportion is greater than 0.3.
2. b) Using a normal approximation to $B(50, 0.3)$ with continuity correction, find the p-value.
3. c) State and justify your conclusion at the 1% level, and explain what a Type I error would mean here.

A company in Dublin offers an insurance policy for €80 per year. The probability of a claim of €500 is 0.1 and of a claim of €2,000 is 0.02; otherwise no claim is made.

1. a) Find the expected payout and the company's expected profit per policy.
2. b) Find the standard deviation of the payout.
3. c) For 1,000 independent policies, find the expected total profit and the standard deviation of the total profit, and hence comment on the company's risk.

A player pays €$c$ to play a game: a fair coin is tossed until the first head appears, at most 4 tosses. If the first head occurs on toss $k$, the player wins €$2^k$; if no head appears in 4 tosses, the player wins nothing.

1. a) Find the probability distribution of the winnings.
2. b) Find the expected winnings.
3. c) Find the value of $c$ that makes the game fair, and determine whether the game favours the player when $c = 3$.