Beat the Odds

Question

You're given the opportunity to make money by betting a total of €100 on the outcome of two, simultaneous matches.

 	Match 1 is between team A and team B
 	Match 2 is between team C and team D

Furthermore,

 	Team A's probability of winning is 40%
 	Team C's probability of winning is 70%

The bookmaker gives you the following betting odds:

 	Team A - 7:4
 	Team B - 2:3
 	Team C - 1:4
 	Team D - 3:1

How much money do you bet on each team? You do not have to bet all €100, but your bets must be whole numbers and the total of all five blanks (bets on the four teams and the unbet amount) must sum to €100. There is no single "correct" answer, but there are many "wrong" answers.

Note: a hypothetical team having 2:7 odds means that if you bet €7 on that team and they win, you get your €7 bet back and win an additional €2. If that team loses, you lose your €7.

Team A bet: ______
Team B bet: ______
Team C bet: ______
Team D bet: ______
Unbet amount: ______

Accepted Answer

Answer with guaranteed profit
The following approach results in finding an arbitrage opportunity.

Match 1:

 	Team A has a payout at success of (7+4)/4 = 2.75. The probability of winning is 40%, so the expected value of the payout is 0.4*2.75 = 1.1 (or, 110%, so you're expected to win money).
 	Team B has a payout at success of (2+3)/3 = 1.67. The probability of winning is 60% (= 100% - 40%), so the expected value of the payout is 0.6*1.67 = 1 (or, 100%, so you're expected to neither win or lose money)).

Match 2:

 	Team C has a payout at success of (1+4)/4 = 1.25. The probability of winning is 70%, so the expected value of the payout is 0.7*1.25 = 0.875 (or, 87.5%, so you're expected to lose money).
 	Team D has a payout at success of (3+1)/1 = 4.00. The probability of winning is 30% (= 100% - 70%), so the expected value of the payout is 0.3*4.00 = 1.2 (or, 120%, so you're expected to win money).

In match 1, the expected payout for both options is 100% or higher, which gives us the suspicion that there is an arbitrage opportunity. To get a payout of at least €100 in this game, we need to bet at least:

 	100/2.75 = 36.36 on Team A
 	100/1.67 = 60.00 on Team B.

These numbers don't sum up to 100, so we're able to bet more on them, for example:

 	If we bet €38 on Team A, our payout is €104.50.
 	If we bet €62 on Team B, our payout is €103.33.

In other words, we can use our full budget of €100 on Match 1 only, and always win.

Alternative Approach
You can also compare the odds with the probability of winning.

Match 1:

 	Team A - Probability of winning is 40% - Ratio of the probabilities in this game is 60/40 = 1.5 (or 150%)
 	Team B - Probability of winning is 60% - Ratio of the probabilities in this game is 40/60 = 0.67 (or 67%)

Next, deduct the payout ratio from the probability ratio:

 	Team A has a payout ratio of 7/4 = 175%. Now, 175-150 = 25%, which is positive. Therefore, the payout for Team A seems higher proportionally to the odds of actually winning.
 	Team B has a payout ratio of 2/3 = 67%.  Now, 67-67 = 0%. Therefore, the payout for Team B as provided by the bookmakers seems fair and correctly priced.

Match 2:

 	Team C - Probability of winning is 70% - Ratio of the probabilities in this game is 30/70 = 0.43 (or 43%)
 	Team D - Probability of winning is 30% - Ratio of the probabilities in this game is 70/30 = 2.33 (or 233%)

Next, deduct the payout ratio from the probability ratio:

 	Team C has a payout ratio of 1/4 = 25%. Now, 25-43 = -18%, which is negative. Therefore, the payout for Team A seems lower proportionally to the odds of actually winning.
 	Team D has a payout ratio of 3/1 = 300%.  Now, 300-233 = 67%, which is positive. Therefore, the payout for Team D seems higher proportionally to the odds of actually winning.

From this approach, it seems attractive to bet on Team A or D, since those odds seem to overpay compared to their actual odds of winning. Betting on Team C seems like a bad idea, since the payout is less than it 'should'. You can come up with different reasonings to how to split the money over Team A or D. For example, you could split it arbitrarily by checking the ratio of overpaying for both teams: 0.67/0.25 = 2.67, which is roughly equal to 73/27. So, bet €73 on Team D, and €27 on Team A.

Title	Category	Subcategory	Difficulty	Status
Parimutuel Betting	Probability and Statistics	Game Puzzle	Medium	Unlock
Tennis Odds	Probability and Statistics	Game Puzzle	Hard	Unlock

The Interview Guide for Quants and Traders

Beat the Odds

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