Sixty-four teams are playing a knockout tournament, and you have to predict the winners of each game. You get 1 point for correctly predicting the first-round winners, 2 for the second, 4 for the third, 8 for the fourth, 16 for the semi-finals and 32 for the final. If you flip coins to choose the winners, what is the expected number of points for predicting all the matches?
Round 1
The expected value of an event is the probability of occurrence x the payout. For a first-round match, the probability of predicting a single winner is (1/2), and the payout is 1.
The expected value for a single first round match = (1/2) x 1 = 1/2
Since there are 32 matches, the total expected value = 32 x 1/2 = 16
Round 2
The probability of predicting a winner in the second round means you need to get two coin flips right (first round and second round). The probability is (1/2) x (1/2). The Payoff is 2. The expected value is (1/2) x (1/2) x 2 = 1/2. For all the 16 matches in this round, it is 16 x 1/2 = 8.
Round 3
The probability is (1/2) x (1/2) (1/2). The Payoff is 4. The expected value is (1/2) x (1/2) x (1/2) x 4 = 1/2. Total = 8 x 1/2 = 4.
The next 3 rounds
The expected values for the next three rounds are 4 x 1/2, 2 x 1/2, and 1/2, respectively.
Adding all values: 16 + 8 + 4 + 2 + 1 + 1/2 = 31.5.
