Quick math probability question

1. May 18, 2005

seiferseph

An unbiased coin is flipped twice. What is the probability that the second one is heads, given the first was a head?

also, is this the same question

An unbiased coin is flipped twice. What is the probability that they are both heads, given the first was a head?

i get 1/2, but i'm not sure if that is correct, thanks for any help!

thanks!

2. May 18, 2005

arildno

Sure, it is the same question, and yes, the answer is 1/2.

This is an exercise in conditional probability, so you ought to solve it by means of that (I hope that's what you did).

3. May 18, 2005

seiferseph

ok, let me confirm how i did it

sample space is (HH, HT, TH, TT). becuase the first one was heads, TH and TT are eliminated, so the probabilty of getting a head is just HH out of HH and HT, so 1/2. correctly done?

4. May 18, 2005

arildno

Sure, that elimination is what "conditional probability" effects.

5. May 18, 2005

seiferseph

ok thank you, now another very similar problem

An unbiased coin is flipped twice. What is the probability that they are both heads, given one of them was a head?

this eliminates only TT, so would the probability be 1/3 ?

6. May 18, 2005

arildno

Careful now!
"An unbiased coin is flipped twice. What is the probability that they are both heads, given that AT LEAST one of them was a head?"

7. May 18, 2005

seiferseph

yeah, then what is the answer to
An unbiased coin is flipped twice. What is the probability that they are both heads, given one of them was a head?

what is the difference?

8. May 18, 2005

arildno

Erm, that was stupid of me. You were right..

9. May 18, 2005

seiferseph

actually you're right, i don't think what i posted was ever printed in a question. they all use "AT LEAST". thanks for the help

10. May 19, 2005

honestrosewater

I think arildno made a good point. Unless you specify "at least", "at most", or "exactly", I think it's customary to assume "exactly". Better safe than sorry :)

11. May 19, 2005

AKG

"exactly" would be a bad assumption. The question would then read, "what is the probability that there are two heads given that there is exactly one head?"

12. May 19, 2005

arildno

This is why I found my objection dumb.
It is a meaningful question whose answer should be 0, but that wasn't what I had in mind when I posted by dumb objection..

13. May 19, 2005

honestrosewater

Right, not really worth asking, but "at most" isn't so far-fetched. I was just saying it wasn't a dumb thing to point out (and I would imagine a rather automatic response in a mathematician or scientist) even if it's quite certain in this case what was meant. It's still better to not need to make an assumption.

14. May 19, 2005

arildno

Agreed; it was more of an automatic response from ingrained habit than the result of deep thinking; but perhaps it is best to develop some such habits after all..

15. May 19, 2005

honestrosewater

Actually, AKG's right, I should have said it's customary to assume "exactly" unless the context suggests otherwise. Just think of how many times you mean "exactly" but don't state it compared to how many times you mean "at least" or "at most" but don't state it. But okay, I'll shut up about it now. :)