Question

Two players take turns tossing two coins. Player 1 scores a point when both coins are the same, that is either a head/head or tail/tail. Player 2 scores a point when the two coins are different, that is either a head/tail or a tail/head. Determine whether the game is fair. Explain why or why not.

Answer 1

Probability

When tossing two coins, four events can happen.

Let's identify as H when the coin shows head and T when shows Tail

The four possible events are shown below:

E={HH,HT,TH,TT}

The game will be fair if both players have the same probability to score points, so we have to calculate their probabilities.

The first coin has a probability of 0.5 of showing H and 0.5 of showing T.

If we have H, then the second coin has exactly the same probabilities, thus we can have HH or HT and the probability of each one is 0.5*0.5 = 0.25

The same happens if the first coin gives T, the two combinations TH and TT have an individual probability of 0.25.

Thus the probabilities of the above events is:

P(E) = {0.25, 0.25, 0.25, 0.25}

Player 1 has a total probability of 0.25+0.25 = 0.5 to score

Player 2 has a total probability of 0.25+0.25 = 0.5 to score

Thus the game is fair because both players have the same probability to score (50/50)