Probability Calculation
At lunch the other day, we noticed that of the five of us at the table, three were left handed (which of course led to complicated seating scenarios). But it also led us to wonder the probability of such an event, assuming that 90% of the world is right handed.
To the rescue comes a modified Pascal's triangle, or more precisely, the coefficient expansion of (9x + y) ^ 5. The neat thing about the solution is that it generalizes so easily - if it turns out that 80% of people are right handed and you want to know the odds of 4 left handed people out of 6, simply look at the appropriate coefficient in (8x + 2y) ^ 6.
Returning to our problem, the final expansion was:
59049x^5 + 32805x^4y + 7290x^3y^2 + 810x^2y^3 + 45xy^4 + 1y^5
The total of the coefficients is a convenient 100,000 and therefore the probability of three left handed people (y^3) is 810/100000 or 0.81%.
Now we can return to the business of eating lunch.