# Odds of getting NO Powerball lottery numbers right?

Yesterday I thought I’d win \$600 million, so I played the Powerball lottery. Not a single number matched up with the winning numbers! What are the odds of this happening? (I’m not even sure how many numbers there are to choose from.)

I’m not particularly good at probabilities, but I can at least get us started. There are five white balls chosen out of 59, and the order isn’t important, and one red ball chosen out of 35. Thus there are 59C5 or 5,006,386 different combinations of white balls, and 35C1 or 35 different red balls. There’s obviously a 34 out of 35 chance of getting the red ball wrong. Now we just need to calculate how many of the 5,006,386 ways of choosing white balls don’t have any of the five winning numbers.

Powerball lists odds of 1 in 31.85 for winning any prize. That’s a 96.86% chance of not winning. But you can get one or two white balls right and still not win. So the odds you’re looking for are going to be somewhat less than that.

Forgive me if I’m misunderstanding how Powerball works; I’m just going by what Wikipedia says: “The game uses a 5/59 (white balls) + 1/35 (Powerballs).

There are [sub]59[/sub]C[sub]5[/sub] = 5,006,386 ways of picking 5 white ball numbers. Of these, [sub]54[/sub]C[sub]5[/sub] = 3,162,510 have no matches with the 5 chosen numbers, because this counts the number of ways of selecting 5 numbers from among the 54 that were not chosen. So the chance of no white ball numbers correct is 3162510/5006386 = 0.6317. The chance of doing this and also getting the Powerball number wrong is this number x 34/35, or 0.6136.

So, well over 50%.

What you want to look at is the odds of NOT picking a ball.

So P(No white balls) = (54/59) * (53/58) * (52/57) * (51/56) * (50/55), and P (No red ball) =34/35). Multiply the two together to get your answer of 61.3%

I’m surprised the answer is so high. I was thinking of that question that asks, how many people do you need in order to have an odds-on chance that at least two of them will share a birthday? The answer is surprisingly low, so I thought the same thing would apply to my lottery example.

The number of balls needed seems surprisingly low to me to the same degree as the birthday paradox. 23 vs 365, 6 vs 55.

I got 10 quick-picks and did not match a single number amid the 60. I think that’s a record.

(Even as hard a mark as I am, I get suckered for 5 or 10 QPs every time the jackpot goes into the stratosphere. Oh, well.)

I got two tickets. One in the morning and then one again in the evening because apparently my close relative didn’t think I’d have enough spare cash to spread around if I won 600 million:rolleyes:

I thought it interesting that two different random tickets purchased at different places at different times had 3 numbers that were the same.

Texas Lotto now has a game called “All or Nothing” wherein you can win the jackpot by matching all 12 numbers or by matching no numbers out of 12.

Is it called Lotterous?

The lottery for the rest of us?

I played reverse Bingo once. The winner was the “last man standing” that hadn’t yet matched a single number. An interesting twist, I thought.

Og nib?

No, Og smash.

Probably not a record, but pretty improbable at a 0.757% chance.

You have to pick 12 numbers from 1 to 24. The odds of matching all 12 or matching none are identical.

Personally, the odds seem to be about 95%.

I meant a record for me. The odds are small but not negligible - I hadn’t calculated them but figured around 1%.