This is not related to winning the lottery.
Instead I had an unusual experience with a lottery ticket I purchased a few weeks ago. I bought ten quick-pick lottery tickets all on one piece of paper.
I noticed that 5 out of 10 tickets had the same Powerball number.
That seems incredibly low likelihood, but I can figure out how to calculate those odds.
If we ignore the other numbers, there are 26 Powerball numbers.
What are the odds that 5/10 randomly chosen numbers out of 26 would be the same?
10c5*(1/26)^5*(25/26)^5 = 0.000017
Actually that’s not right… that would be the probability that 5 of the 10 would be a particular number, that 5 of the the 10 are the same number is about 26 times more likely, so around 0.0005.
I am less clear on the question than the unwashed one seems to be.
So you have ten rolls of a 26-sided die, and five of them came up with the same number?
Essentially, yes.
NM. Not enough coffee yet. Answer above seems too complex.
That’s what I took it to mean.
If we ask the question what is the probability of rolling exactly five 7s (for example), the answer is given by
10C5 * (1/26)^5 * (25/26)^5 = 0.000017
because that’s just a regular binomial distribution with N=10; P(S) = 1/26 and K = 5
But that wasn’t the question, which was what is the probability of rolling exactly 5 of no particular number? i.e. rolling exactly five 1s or exactly five 2s or exactly five 3s or etc
The reason why it’s not exactly 26 * 10C5*(1/26)^5*(25/26)^5 is because some outcomes have five of one number and five of another. But the probability of each of those outcomes is order of magnitudes smaller than out original answer, so for good approximation purposes we can ignore them.
I guess my confusion is outlined above - is the OP’s question “five of the same number” or “five of the same specific number”? I took it to be the run of five numbers, which could have been 10 or any other possible number.
(my coloring)
It’s the green question.
I don’t follow what you say in red, there’s no requirement that the hits be consecutive (in a “run”).
Sorry, I am expressing myself poorly here.
I believe the OP is asking what the odds are of receiving five of the same number among ten - any same number from 1 to 26 - and not the odds of receiving five of a specific number (10) among those ten.
We agree! Yay!
I’ll even overlook that you edited my quote. 
Whether it’s the probability of EXACTLY 5 of 10 tickets or AT LEAST 5 of 10 tickets makes a difference. One is more likely than the other, though not by a huge amount. After all, if it was 7 of the same Powerball number, the same question would likely be asked.
It becomes a basic binomial probability problem.
For EXACTLY 5 draws (already answered above), there are C(26,1)=26 ways of picking which of the 26 numbers gets picked 5 times, and C(10,5) ways of picking which 5 of the 10 tickets these are. There is a (1/26)^5 chance of picking the powerball 5 times and a (25/26)^5 chance of not picking it 5 times.
Combining those, we have a probability of:
C(26,1)C(10,5)(1/26)^5*(25/26)^5 = 0.000453 = 0.045%
That’s about 1 in 2200, so not great but not terrible, either.
For AT LEAST 5 draws, we can just add the probabilities for more draws:
C(26,1)C(10,6)(1/26)^6*(25/26)^6 = 0.0000140
C(26,1)C(10,7)(1/26)^7*(25/26)^7 = 0.000000295
C(26,1)C(10,8)(1/26)^8*(25/26)^8 =0.00000000409
C(26,1)C(10,9)(1/26)^9*(25/26)^9 =0.0000000000336
C(26,1)C(10,10)(1/26)^10*(25/26)^10 = 0.000000000000124
Adding these together, the probability of AT LEAST 5 of the same is a bit higher at 0.0467%, or roughly 1 in 2140.
For reference, the probability that all 10 are unique powerballs is reasonably high. There are 26^10 = 1.4110^14 ways of picking 10 powerballs, of which P(26,10)=1.9310^13 will contain 10 unique numbers. That’s a probability of 13.65%.
Thinking more on it, I suppose the analysis above also includes the probability that 5 of the powerballs are one number and 5 are another number, which is a very low probability event but included.
Awesome thanks!
So even though I didn’t win a bazillion dollars I should have kept that ticket because
“Damn! That’s unusual!”
Ouch. I realized my numbers are all wrong for the “AT LEAST 5” cases. Not by much but still…
Corrected numbers:
For AT LEAST 5 draws, we can just add the probabilities for more draws:
C(26,1)C(10,6)(1/26)^6*(25/26)^4 = 0.0000151
C(26,1)C(10,7)(1/26)^7*(25/26)^3 = 0.000000345
C(26,1)C(10,8)(1/26)^8*(25/26)^2 =0.00000000518
C(26,1)C(10,9)(1/26)^9*(25/26)^1 =0.0000000000460
C(26,1)C(10,10)(1/26)^10*(25/26)^0 = 0.000000000000184
Adding these together, the probability of AT LEAST 5 of the same is a bit higher at 0.0465%, or roughly 1 in 2150.
I think this is the tidbit that would surprise most people, especially the statistically / combinatorily naïve.
If you buy a 10-chance ticket, you have a 13.65% ~= 1/7 chance of all 10 powerballs being different. Therefore a 86.35% ~= 6/7 chance of at least one duplication amongst the 10.
The math is unassailable, but I suspect a lot of folks would think it flukey, or at least disappointing, to receive a ticket with duplicates amongst the 10.
This is essentially the somewhat counterintuitive “birthday problem”, right?
Yes, exactly the same issue but with 26 instead of 30/31 possibilities. It’s only counterintuitive because untrained human intuition is so bad at math.
It’s called innumeracy (similar to illiteracy).
And it’s the reason lotteries exist at all.
You mean “26 instead of 365/366 possibilities”, at least how it’s usually presented. The question is basically, how many people do you need in a room before it’s more likely than not that two share the same birthday? It’s 23, and you’re choosing from a pool of 365 or 366 days.