Gambling Winning and Losing Streaks, and the Standard Deviation Here is an Excel spreadsheet set up to simulate an even-money coin flip game — heads wins one dollar and tails loses one dollar. The graph shows the change in bankroll for 100 flips. If you were to flip a real coin and have a friend record and graph the results, it would take you about 15 minutes for 100 flips. But here I can press the F9 key and get a new set of results in one second. Based on statistics, I can predict what’s likely to happen in future sessions.
First of all, because this is a fair game, I expect the numbers of winning and losing sessions to be about equal. Now look at these horizontal lines. I’ve spaced them at 10 units, which is the standard deviation for the win or loss after 100 coin flips. It’s like an “average distance” away from the single most likely result, an exact match of 50 wins and 50 losses.
The final result will end up within one standard deviation, between HERE and HERE, about 2/3 of the time; and within two standard deviations, between HERE and HERE, 95% of the time, and within 3 standard deviations, somewhere on this graph, almost always, 99.7% of the time. I’ll press the F9 key 20 times to generate 20 sessions of 100 flips each, or 2000 flips all together. I expect about 13 sessions to end up within one standard deviation, and about 19 out of 20 sessions within 2 standard deviations, with only about 1 session outside of that range, and none ending completely off the graph, more than 3 standard deviations.
I also predict we’ll see a winning streak of 10 or more consecutive wins without a single loss, AND a losing streak of 10 or more consecutive losses without a single win. If fact, I would not be surprised to see multiple instances of these streaks. OK, here we go, 20 sessions.
1, 2, ooh, that’s a big deviation. I expect to see this once out of 20 times. And here, is that a 10, 10 consecutive winning streak? From point 73 to 82. Almost!
Let’s see, was than number 3? Here’s number 4, 5, 6, 7, 8, 9, 10. Here’s our big losing streak, more than 10, let’s see. From point 55 to point 65, exactly 10. I forgot where we were.
Was that 13? Here’s your big losing streak, right here. 14, 15, 16, 17, 18, 19, and 20. Later on I’ll tally the results in the description section. If you would like to play around with this simulation yourself, I have another video that shows you how to set it up in Excel, step by step. You can have it up and running in 5 or 10 minutes.
So what about sessions shorter or longer than 100 flips? This chart is a summary. For a session of this number of flips, you have a standard deviation in your win or loss of this much. The average final result is break even, as shown here.
A moderately UNLUCKY final result, one standard deviation down, is this much; and a moderately LUCKY final result, one standard deviation up, is this much. In 2/3 of the sessions, you can expect to end up somewhere between these results, and 1/3 of the time, something more extreme. Because this is a fair coin flip game, the chance of a positive result is 50%.
As the sessions get longer and longer, the standard deviation gets bigger and bigger, but the deviation as a fraction of the number of expected wins gets smaller and smaller, as indictated in this column. For example, for 50 flips, you can expect to win about 25 times, with a standard variation of about 7%. But for 2,500 flips, you can expect to win about half of that, 1,250 times, with a variation of just 1%. The longer you play, the closer you will end up with the expected results, fraction-wise. This becomes really important for UNFAIR games, when you have a statistical disadvantage.