1. ## Probability question

You and your other friend, B, owes C an amount of money (X). But instead of just asking you to pay him the money, C wants to play a game. If you lose you give him the double amount, but if you win you don't have to pay at all. The game consists of that you draw balls from an urn which contains 18 balls, of which 12 are white and 6 black. If you draw a black one, you lose. If you draw a white ball, you win.

I first define the events:

A = I win
B = B wins

a) If you draw the first ball, what is the probability that you win given that B also wins?
We must be looking for the union P(A union B) = P (A) * P(B given A) = (12/18 * 11/17). Is this correct?
b) Is it a good deal for you to accept the offer from C, instead of just paying him the money?
I guess I must calculate some kind of expected value for how much I will gain from the game. If I define Y = win and Z = loss, then E(gain) = E(Y) - E(Z). Is this the right way to think? But how do I calculate these two values?

Thanks a lot for for your help!

2. Originally Posted by mirrormirror
You and your other friend, B, owes C an amount of money (X). But instead of just asking you to pay him the money, C wants to play a game. If you lose you give him the double amount, but if you win you don't have to pay at all. The game consists of that you draw balls from an urn which contains 18 balls, of which 12 are white and 6 black. If you draw a black one, you lose. If you draw a white ball, you win.

I first define the events:

A = I win
B = B wins

a) If you draw the first ball, what is the probability that you win given that B also wins?
We must be looking for the union P(A union B) = P (A) * P(B given A) = (12/18 * 11/17). Is this correct?
b) Is it a good deal for you to accept the offer from C, instead of just paying him the money?
I guess I must calculate some kind of expected value for how much I will gain from the game. If I define Y = win and Z = loss, then E(gain) = E(Y) - E(Z). Is this the right way to think? But how do I calculate these two values?

Thanks a lot for for your help!
I suggest drawing a tree diagram. From this you will see that

a) $\Pr(A \, | \, B) = \frac{ \left( \frac{12}{18}\right) \cdot \left( \frac{11}{17}\right) }{\left( \frac{12}{18}\right) \cdot \left( \frac{11}{17}\right) + \left( \frac{6}{18}\right) \cdot \left( \frac{12}{17}\right)} = \, ....$

b) Let the amount owed be $x$. Suppose A pays the friend $x$. Now, if A wins the friend gives A $x$, if A loses A gives the friend another $x$.

Calculate the expected winnings of A for the game: $E(W) = \Pr(A) \cdot x + \Pr(A') \cdot (-x)$.

If E(W) > 0 it's a good deal. If E(W) = 0 it doesn't matter whether A plays or not. If E(W) < 0 it's a bad deal.

3. Thanks a lot! But I guess that 6/18 in the denominator is a typo? Shouldn't it be 12/18, or am I wrong?

4. Originally Posted by mirrormirror
Thanks a lot! But I guess that 6/18 in the denominator is a typo? Shouldn't it be 12/18, or am I wrong?
You're wrong.

The denominator is Pr(B) = Pr(B | A) Pr(A) + Pr(B | A') Pr(A') ....