Implied Probability

Section E.2 Implied Probability

More generally, we can compute the expected value of each wager for a range of probabilities from 0 to 1.

Listing E.2.1. Python Code

ps = np.linspace(0, 1)
ev_patriots = expected_value(ps, 100, 195)

Listing E.2.2. Python Code

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ps = np.linspace(0, 1)
ev_seahawks = expected_value(1-ps, 240, 100)

Here’s what they look like.

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Listing E.2.3. Python Code

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plt.plot(ps, ev_patriots, label='Bet on Patriots')
plt.plot(ps, ev_seahawks, label='Bet on Seahawks')
plt.axhline(0, color='gray', alpha=0.4)

decorate(xlabel='Actual probability Patriots win',
        ylabel='Expected value of wager')

To find the crossover point, we can set the expected value to 0 and solve for p. This function computes the result:

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Listing E.2.5. Python Code

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def crossover(wager, payout):
    return wager / (wager + payout)

Here’s the crossover for a bet on the Patriots at the offered odds.

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Listing E.2.6. Python Code

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$ p1 = crossover(100, 195)
p1
0.3389830508474576

If you think the Patriots have a probability higher than the crossover, the first bet has positive expected value. And here’s the crossover for a bet on the Seahawks.

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Listing E.2.7. Python Code

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$ p2 = crossover(240, 100)
p2
0.7058823529411765

If you think the Seahawks have a probability higher than this crossover, the second bet has positive expected value.

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So the offered odds imply that the consensus view of the betting market is that the Patriots have a 33.9% chance of winning and the Seahawks have a 70.6% chance. But you might notice that the sum of those probabilities exceeds 1.

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Listing E.2.8. Python Code

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$ p1 + p2
1.0448654037886342

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