re: Flip a coin 6 times. What are the odds for exactly 5 tails?

7

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I think the odds for getting any particular distribution you want to pick are always going to be the same: 0.5^6

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Any single distribution, yes. But OP asked for exactly 5 tails. There are 6 ways to roll that:

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HTTTTT
THTTTT
TTHTTT
TTTHTT
TTTTHT
TTTTTH

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And 2^6=64 unique rolls

6/64=9.375% probability of exactly 5 tails

But he asked for the odds, so scenarios where it happens:scenarios where it doesn't.

(6/64)/(1-(6/64))
(6/64)/(58/64)
(6/58)=(3/29)
Like I said before, it's counterintuitive to write odds as a fraction in that way because it's too easily confused with probability. I'd write it as 29/3 or 29:3 if this was gambling, which is the same as 9.375% probability

ETA: the way I did 6/64 is the intuitive way, but for larger samples (say, flipping a coin 1000 times and wanting the probability of exactly 746 tails) you'd use the equation for binomial distribution. Unless you really like writing out permutations by hand

Yup, it just struck me and I was coming back in here to edit my post

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But he asked for the odds

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That was on purpose but not as a gotcha. Until my son came to me with this problem from his LSU probability course I've always conflated probability with odds like most people. The two concepts are different as you have outlined.

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0.5^6

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That's the probability of six heads in a row which isn't what the OP needs.