I'm trying to calculate the probability of winning powerball.

Question

Using conditional probability without replacement I find that the chance of getting 5 white balls (69 uniquely labeled balls in total) correct is 1/1348621560. The problem is when I use binomial coefficients to find the number of possible coefficients I end up with a different probability of 1/11238513. Why do I get 2 different probabilities? I think I got confused somewhere can someone help?

Answer 1

Hi Victor! 

 

I am not sure exactly how powerball is set up, but I am assuming that your numbers just have to match the five numbers drawn by the power ball people. In other words, we are choosing a set 5 distinct numbers from 69 distinct numbers. The order of the draws does not matter. This is 69 choose 5 = 11238513. So this way of thinking aligns with your second answer.

 

However, if the order of the balls DOES matter, then we will get a larger denominator. Namely we use 69!/64!.