A slippery subject
Probability is a slippery subject in more ways than one.
Here's an example (from Martin Sternstein's Statistics Barron's, 1994) of a possibly counterintuitive result:
Suppose that 6 percent of stories of major stock market success stem from illegal insider information passing. In some arbitrary group of seven highly successful stock traders, what would you think would be the chance that only one member of the group is a crook whose success stems from illegal insider activity?
The answer may surprise you.
We apply the binomial formula C(n,x)(p^x)(q^n-x). That is, 7!/6!(0.06)(0.94)^6 = 0.2897. So, the chance only one member is a crook is close to 30 percent!
The chance that at least one member is a crook is 1-(0.94)^7 = 0.351, or about 35 percent. That is, the chance is greater than one in three that one successful member of some group of seven successful stock traders is a crook -- despite the overall low incidence of criminality!