We did a little impromptu talk about the math behind Powerball this morning. We also did a similar talk last summer:

Playing around with the odds of winning Powerball

but I learned this morning that the game has changed slightly since our original talk, so it seemed worthwhile revisiting the topic of Powerball odds today.

The official listing of the odds for each prize in Powerball is here:

The official odds of winning various prizes in Powerball

The specific topics for today were:

(1) What is the probability of winning the grand prize?

(2) What is the probability of getting only the Powerball correct?

(3) Since the chance of getting winning the grand prize is about 1 in 292 million, what is the probability that no one will win if 292 million people pick numbers at random?

The third part was a little hard for the kids to understand than I was expecting it to be, but the extra time we spent talking about this question made the project worthwhile.

Here’s the first part – the chance of winning the grand prize:

The next part was about the chance of getting only the Powerball correct. The FAQ on the Powerball site has a funny comment about this particular prize.

Finally, the talk about no one winning if 292 numbers were guessed took three parts. Here’s the introduction to that question. The kids were a little confused about how to approach it.

I tried to simplify the problem by looking at a case where 2 people guess at a number that is either 0 or 1:

After looking at the easier version of the problem we returned to the problem of no one winning the grand prize in Powerball if 292 million people played. It was sill difficult for my younger son to understand this problem, but we did eventually get to the punchline – the probability is approximately 1 / e.

Where did e come from??

So, a fun project. I’m guessing there will be a lot of conversations about Powerball over the next few days – maybe that’ll get kids interested in talking about the math behind the game, too.