Poisson Distribution

p(4T~10T) = 0.1428, or almost exactly 1 chance in 7.

Explanation

Let's express the problem in more typical Poisson terms, which means in terms of the rare event, which here is Tails. The rate r for Tails with Rick's crooked coin is p(T) = 0.2000, so that the likeliest outcome is 2 tails per 10 tosses. What we want to find out is, what is Jimmie's chance of winning? By definition, Rick wins if there are 7, 8, 9, or 10 Heads in the 10 tosses. That is, he wins if there are 3, 2, 1, or 0 Tails. Then Jimmie wins if there are 4 or more Tails, and his chance of winning is the sum of the probabilities corresponding to p(4T~10T) inclusive. For r = 0.2, we find that p(10) is so small that it is not even given on the Poisson Table, but the sum of the probabilities from p(4) to p(9) inclusive is 0.1428, or 1 out of 7. The chances, then, are 6 out of 7 that Jimmie will lose his horse, and walk home from the wager.

Comment

Jimmie is a born fool, and will never buy and sell for a profit. Before gambling with a man, look that man in the eye. If he cannot meet your gaze, he is a crook. If he meets your gaze frankly and fairly, he is a practiced crook. There are no other options. So much for the family horse. The only thing that protects the family farm is that the family kid can't saddle it up and ride it to the county fair.

And in the present age of absentee banking and online refinancing, it would be another fool who would count on even that.