Say that you flip a coin 6 times the probabilities would be broken up like this


Odds of getting no heads = 1 / 64
Odds of getting one heads = 6/64
Odds of getting two heads = (6! / 4! * 2!) / (64) = 15/64
Odds of getting three heads = (6! / 3! * 3! / (64) = 20 / 64
Odds of getting four heads = (6! / 4! * 2!) / (64) = 15/64
Odds of getting five heads = 6/64
Odds of getting six heads = 1 /64

Proof - 1/64 + 6/64 + 15/64 + 20/64 + 15/64 + 6/64 + 1/64 = 64/64 = 1

My question is that intuitively the probability of getting 3/6 tosses as heads would be 50% but it is 20/64 or 31.25%? - How does that work - Or am I doing something wrong??

Please help - Thanks!





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first off, all the probabilities you've listed are correct. so, on to this:



as i noted above, you're not doing wrong, but you're thinking wrong.

here's an analogy:
let's say you toss the coin a million times instead of 6 times.
now, it's true that, of all the different possibilities that can occur, the single most likely is exactly 500,000 heads out of a million tosses.
BUT
what do you think is the actual probability of getting exactly 500,000 heads on a million tosses of a coin?
it's vanishingly small, of course; it would be nothing short of absurd to think it's 50 percent.

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in any case, the realization you've got to make is that, the more tosses you make, the more possibilities there are going to be, and, therefore, the probability of each of those individual possibilities will decrease. it's true that the most likely outcome is still 50% heads, but, when there are hundreds of thousands of possibilities, the most likely outcome is still rather unlikely.

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i think part of the confusion here stems from the fact that, if you flip the coin only twice (one of the most commonly considered cases in elementary treatments of probability), the probability of getting exactly 1 head is, coincidentally, 50%. this fact may well lead you into making false generalizations.

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finally, if you need just one more death blow to your original intuition, think about an odd number of flips. if you flip a coin seven times, do you anticipate that the probability of getting exactly 3.5 heads is 50%? uh...