Why is the probability of flipping half or more heads out of 5 coins, == 1/2, instead of 1/8?

Shouldn't P(>=3 heads | 5 coin flips)== 1/2 * 1/2 * 1/2 for the conditional prob of getting a new head after the previous flip yielded heads?

Is it because this doesn't take into account all the potential combinations? If so, how would I work this out mathematically without the shortcut of using symmetry? How can the symmetry be expressed in math?

I feel I'm missing something quite obvious.

  • 2
    $\begingroup$ Hint: binomial distribution $\endgroup$ – zbicyclist Feb 26 '17 at 19:15
  • $\begingroup$ Would you also then say that the chance of "half or more tails" was also 1/8 by similar reasoning? Notice that the two events together cover the whole sample space (and they're mutually exclusive for an odd number of tosses, since nether can be exactly half), so the probability that either one or the other of those two events happens must be 1. But by symmetry of heads and tails the probabilities of those two events must be equal... Clearly then, by elementary reasoning we can obtain the correct answer without any reference to combinations, binomial distributions etc $\endgroup$ – Glen_b Feb 26 '17 at 23:05

Your intuition is correct. It does have to do with combinations. You do not need to take conditional probabilities into account in this case. The fact that your last flip resulted in heads or tails has no realistic effect on the outcome of the next flip. I would approach this problem in the following way. Let's call the number of coin flips X.

$P(X\geq 3) = P(X=3)+P(X=4)+P(X=5)$

As the ordering do not matter, there are ${5 \choose 3}=10$ valid ways of throwing three heads out of 5. So $P(X=3)={5 \choose 3}*0.5^3 * 0.5^2 = \frac{10}{32}$

We can repeat this train of thought for $P(X=4)$ and see that this equals ${5 \choose 4}*0.5^4*0.5= \frac{5}{32}$

For the last option, you have obviously figured out that this equals ${5 \choose 5}*0.5^{5}*0.5^0=\frac{1}{32}$.

Adding these up, we see that $P(X\geq 3)= \frac{16}{32} = \frac{1}{2}$

This is a classic example of a Binomial Distribution. To get you started: https://en.wikipedia.org/wiki/Binomial_distribution. In my opinion, this article does quite a good job at explaining this concept. If something is still unclear, feel free to ask some more!


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