# Quick math probability question

• seiferseph
In summary: 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..

#### 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!

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).

arildno said:
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?

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

arildno said:
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 ?

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?"

arildno said:
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?"

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?

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

arildno said:
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

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 :)

"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?"

AKG said:
"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..

AKG said:
"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.

honestrosewater said:
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..

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. :)

## 1. What is the definition of probability?

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 represents impossibility and 1 represents certainty.

## 2. How do you calculate probability?

To calculate probability, you divide the number of desired outcomes by the total number of possible outcomes. For example, if you want to know the probability of rolling a 6 on a standard six-sided die, you would divide 1 (desired outcome) by 6 (total possible outcomes), giving you a probability of 1/6 or approximately 16.67%.

## 3. What is the difference between theoretical probability and experimental probability?

Theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely. Experimental probability is based on actual observations and may vary from the theoretical probability due to random chance or other factors.

## 4. How do independent and dependent events affect probability?

In independent events, the outcome of one event does not affect the outcome of another event. In this case, the probability of both events occurring together is equal to the product of their individual probabilities. In dependent events, the outcome of one event is affected by the outcome of another event. In this case, the probability of both events occurring together is equal to the product of their individual probabilities, adjusted for the impact of the first event on the second event.

## 5. What is the difference between permutation and combination?

Permutation is the arrangement of a set of objects in a specific order, while combination is the selection of a subset of objects from a larger set without regard to order. Permutation takes into account the number of ways that the objects can be arranged, while combination does not. This means that the number of permutations will always be greater than or equal to the number of combinations for the same set of objects.

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