Question

1. You try to find out whether a coin is fair or has two heads. The coin is tossed three times, after which you will be shown the outcome of all three coin tosses. If you decide to conclude that the coin is fair if it shows heads and tails at least once, and unfair if all three tosses are heads, what is the probability of committing a type I error and a type II error?

Answer 1

Let p denote the probability of the coin landing on heads on a single toss.

We want to test if the coin is fair (null hypothesis) or has two heads (alternative hypothesis), i.e. we are testing:

H₀ : Coin is fair or p = 1/2, vs

H₁: Coin has two heads or p = 1.

Now, we will conclude that the coin is unfair (having 2 heads) if all the three tosses of the coin land on heads.

Now, the probability of committing  type I error is given by:

Now, the probability of committing  type II error is given by:

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