October 27, 2010

Suppose you were in the business of predicting the outcome of a coin toss, where with each flip of the coin, you have a 50% chance of being correct. What are the odds that out of 20 coin tosses, you would correctly call heads or tails exactly 18 times during all those tosses?

Binomial Probability Data
Input Data Values
Total Number of Opportunities [Must be 170 or lower to avoid maxing out the calculator!]
Number of Times the Outcome Goes a Particular Way
Percentage Odds of Outcome Occurring for Each Opportunity [%]

The Odds of That
Calculated Results Values
Percentage Odds of the Outcome Occurring the Entered Number of Times
Odds of the Event Occurring [1 in ...]
Our newest tool is designed to answer the question of just how likely or unlikely it would be for such a thing to happen. Just enter the indicated data in the tool below and we'll work out the probability of such a thing happening by pure chance for a given number of opportunities!

We find that the percentage odds of correctly calling the outcome of 20 coin tosses exactly 18 times by chance is 0.0181%, or rather, the odds are that this exact situation will occur by chance just once in 5518.8 opportunities.

Now for some food for thought. Since January 1871, the percentage probability that the average monthly price of stocks in the S&P 500 will be higher than in the previous month is 56.1%. What is the likelihood that an individual could correctly anticipate that stock prices would either be higher or lower than the average price level recorded in the previous month on 17 out of 19 occasions by chance?

[Answer: It's very unlikely that chance alone explains that happening, but it's also not as improbable as you might think.... ]

### Elsewhere on the Web

We found two really cool tools for doing this kind of math elsewhere on the web:

Richard Lowry presents the detailed calculations and an online calculator for finding binomial probabilities that gets around our tool's limitation of 170 opportunities!

Texas A&M offers a Java application for doing this kind of math that includes graphical output, so you can see where various outcomes might fall on a normal bell curve distribution!

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