Anne has developed a 3% edge in sports betting based on some intelligent math, but she has yet to learn the exact advantage. She wants to bet $ 1 on a team at odds of 1/2. How many bets does she need to make before she knows her edge?
Before we get into the question, let’s familiarise ourselves with what Nate Silver popularised as ‘signal’ and ‘noise’. The signal is what we expect—in simple language, it’s the mean. The noise is the variability, or, in other words, the standard deviation.
The problem mentioned above is another way of stating the number of trials required for Anne to develop the confidence interval (say 95%) that can clearly distinguish the signal (the edge, 0.03) from the noise (the standard deviation). Just a reminder: for a fair bet, the signal (the long-term average) should have been 0, but since Anne has an edge of 0.03, it must be 0.03.
The confidence interval per average trial is given by the following formula. h is the signal, sigma is the standard deviation, and n is the number of trials.
If the odds are 1/2, for a wager of 1, one gets 0.5 or loses 1. Also, the winning probability is,
2/(2+1) = 0.667.
Standard deviation
The squared distance for a win is (0.5 – 0.03)2 and for a loss (-1 – 0.03)2 . The average squared distance (the variance),
The standard deviation is the square root = 0.71
Confidence interval
Now, all we need to do is estimate n such that the term on the right-hand side of the plus/minus equals or less than the signal value.
1.96 x noise / root(n) < signal
1.96/root(n) < signal/noise
n > (1.96/(signal/noise))2
n > (1.96/(0.03/0.71)2
n > 2140
Reference
The Ten Equations that Rule the World: David Sumpter
