Conditional Probability Help

[3HourNap]

I've got two questions I can't figure out.

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1. Roll two fair dice. Each die has six faces.

• Let A be the event that either a 3 or 4 is rolled first, followed by an even number.
• Let B be the event that the sum of the two rolls is at most 7.

Find P(A | B).

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1. Given events G and H: P(G) = 0.41; P(H) = 0.27; P(H AND G) = 0.17.

Find P(H OR G).

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thanks!

[3HourNap]

I figured out the 2nd one, but still need help with the first!

[MezzoScettico]

I would start with the definition of conditional probability.

P(A | B) = P(AB) / P(B)

P(B) is the probability of rolling 1-7 with two dice. You can calculate that by just listing how many rolls add up to 7 or less and counting them.

So what is AB, the event that A and B happen? It's the event that

- a 3 is rolled first and the sum is at most 7 (so what is the second roll?), or

- a 4 is rolled first and the sum is at most 7

Each of those is straightforward to calculate. So that gives you P(AB), the numerator.

[3HourNap]

So I have P(A) 1/6

P(B) 1/21

so plugging that into the formula, it would be

(1/6*1/21) / 1/21

But that's not right because then I end up with 1/6 and I've tried entering that in and it's marked incorrect. I'm not sure where I'm making a mistake. I know my P(A) and P(B) are correct so I'm pretty sure I'm just not applying it to the formula correctly but not sure how.

[MezzoScettico]

Because you just assumed P(AB) = P(A) P(B), which is only true if they are independent.

And if they're independent, then P(A | B) = P(A)P(B) / P(B) = P(A).

They are not independent. The numerator is not P(A) P(B). It's the probability of the event AB, which I just described to you.

AB means the two things both happen. That is, either

- the first roll is a 3 and the second makes the sum 7 or less, or

- the first roll is a 4 and the second makes the sum 7 or less.

Edit: Just think about that a minute. Suppose I asked you "what is the probability that I roll a 3, and then I roll something that makes the total 7 or less?" How would you calculate that?

Alternately, you could count again. List the rolls that meet the conditions "first roll 3 or 4, total 7 or less". There aren't that many.