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

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