After 10 heads, what is the chance of tails next?

Question

I was arguing with my friend about this and we need a mathematician to settle it.

I flipped a coin 10 times and got heads every single time. My friend says "the next flip is MORE likely to be tails because its due."

I said "no, the coin doesnt have memory, its always 50/50."

But then my friend said "if its always 50/50, then the probability of 11 heads in a row is $(0.5)^{11} \approx 0.0005$ , which is basically impossible. So if ive already seen 10 heads, the 11th being heads would mean an extremely unlikely event happened, so its more likely to be tails."

Is this reasoning correct?? It feels wrong but I cant explain why mathematically. Something about conditional probability vs absolute probability.

Also: if I HADNT observed the 10 heads, then 11 heads in a row is indeed $(0.5)^{11}$ . But since ive already observed 10 heads, the conditional probability $P(\text{11th heads} \mid \text{10 heads})$ is just $0.5$ . Is that right?

Kenji Nakamura · Accepted Answer

If the coin is fair and each flip is independent, the probability of tails on the 11th flip is still: 1 2 . The first 10 heads are surprising as a sequence, but they do not change the physical coin. The mistake is thinking tails is "due." That is the gambler fallacy. Independence means: P ( tails on flip 11 ∣ 10 heads before ) = P ( tails on flip 11 ) = 1 2 . What is unlikely is the whole sequence: H H H H H H H H H H T . That specific 11-flip sequence has probability: ( 1 2 ) 11 . But once the first 10 flips have already happened, you are only predicting the next flip. For a fair coin, that next flip is 50 percent heads and 50 percent tails.

After 10 heads, what is the chance of tails next?

1 Answer

After 10 heads, what is the chance of tails next?

1 Answer