Quick math probability question

[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!

 

[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).

 

[seiferseph]

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).

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?

 

[arildno]

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

 

[seiferseph]

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

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 ?

 

[arildno]

Careful now!
Wouldn't you say that your answer is the answer to the question:
"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?"

 

[seiferseph]

Careful now!
Wouldn't you say that your answer is the answer to the question:
"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?"

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?

 

[arildno]

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

 

[seiferseph]

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

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

 

[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 :)

 

[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?"

 

[arildno]

"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?"

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..

 

[honestrosewater]

"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?"

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.

 

[arildno]

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.

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..

 

[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. :)