Super Bowl

Data and Decisions Case nº 1: The Super Bowl                               

Question 1

We will check if the probability of winning obtained assuming a fair bet meets the three rules of probability. All the events obtained are a partition of the sample space as it is not possible that 2 teams win the Super Bowl at the same time so all pairwise are mutually exclusive. In addition the Super Bowl will be won by one of the 32 teams so the events are collectively exhaustive.

1. All probabilities are between 0 and 1: Yes all of them meet this property. The smallest one is ~0.006622517 while the largest one is ~0.181818182

1. P(S)=1 The sample space is made up from all the teams(32) that could win the Super Bowl.  P(S)= 1.325129574. So this key property of probabilities is not met by the table of probability obtained.

1. As events are mutually exclusive the probability of the intersection of any 2 events is 0. So P(AUB) = P(A) + P(B) – P(A   B) = P(A) + P(B). This property is actually met for any couple of events as the probability that one team or another win the Super Bowl is the sum of the probabilities.  [pic 1]

So the basic rules for probabilities are not met. As the sum of the probabilities obtained is higher than 1 that means that there is an opportunity for arbitrage.

Question 2:

To check if there is an arbitrage for the betting houses through a specific sequence of bets we have to calculate the worst case scenario (i.e the case where the customer would have the highest profit) and check if it involves a profit or a loss for the betting house.

• Case a: 32 bets, each of size $1 on each of the thirty two listed teams

In this case the worst case scenario for the betting house would be that Oakland or Tenessee won the SuperBowl. In this case the betting house would have to pay to the player 150$ +1$ of the initial bet. But the player has bet 32$ 31$ in all other bets that he has lost and the 1$ that he had to invest in the winning bet to receive the 151$. So the net loss of the betting house is: 151$-32$=119$. This case does not involve an arbitrage for the betting house as there is at least a case when it would lose money. In fact the betting house will lose money in all cases where the winning team has odds higher than 32$ as in any of those cases the betting house would pay more than it would receive.