# Independent events in sequence

If I flip a fair coin twice, the probability of at least one Head is 0.75 (HH, HT, TH and TT).

I flip the coin once and it lands Tails. In order to get at least one Head, the probability of the second coin flip landing Heads must now be 0.75, but this is not possible, as it is (obviously) 0.5. What am I missing?

• The conditional probability of observing at least one Head given that one Tail was observed is not the same as the unconditional probability of observing at least one Head. Commented Dec 16, 2014 at 13:09
• You indeed forgot to condition. Commented Dec 16, 2014 at 13:25

If I flip a fair coin twice, the probability of at least one Head is 0.75 (HH, HT, TH and TT).

Before you flip at all, the probability you'll get at least one head in two flips of a fair coin is 0.75, yes. This is because the four outcomes ($HH$, $HT$, $TH$ and $TT$) are equally likely.

I flip the coin once and it lands Tails.

At this point, you're no longer in the position you were when you calculated p=0.75

You have additional information, and the probabilities of the final outcome change when the information changes -- the first toss is certain. The coin doesn't adjust to make the final probability 0.75 (how could it?), it's still a 50-50 shot.

So once you have observed a tail, the chance of getting HH or HT are 0. The only possibilities are Tail-something, and the two possibilities for something are equally likely. Once you have an initial T, you can only end up with TH or TT, each with a 50% chance.

So the probability of "at least one head" is reduced when you know the first toss is a tail.