# Homework Help: Expected profits in Blackjack - tricky probability problem

1. May 18, 2014

### Nikitin

1. The problem statement, all variables and given/known data

3. The attempt at a solution

I've done a) $X$~$f(x) = 0.7^{x-1} 0.3$, and b) $W = 2^{X-1} \cdot 2 \Rightarrow E(W) = \Sigma_{x=1}^{5} f(x) 2^x = 6.59$.

I'm completely stuck on c though. I keep getting the wrong answer, as I try to calculate the expected loss. For instance, why is the following wrong?

$E(L) =$
$+ 0.3\cdot (0)$
$+ 0.7 \cdot 0.3 \cdot (1)$
$+ 0.7 \cdot 0.7 \cdot 0.3 \cdot (1+2)$
$+ 0.7 \cdot 0.7 \cdot 0.7 \cdot 0.3 \cdot (1+2+4)$
$+ 0.7 \cdot 0.7 \cdot 0.7 \cdot 0.7 \cdot 0.3 (1+2+4+8)$
$+ 0.7\cdot 0.7 \cdot 0.7 \cdot 0.7 \cdot 0.7 \cdot (1+2+4+8+16)$
$= 7.5$

I mean, these are all the possibilities if Lars is only playing a maximum of 5 times?

Last edited: May 18, 2014
2. May 18, 2014

### Staff: Mentor

Are E(L) the expected bets? Then the first line (Lard wins the first round) should not be zero as he still has to put his bet.
Is E(L) the expected total result? Then the first line (Lard wins the first round) should not be zero as he gains money.

The other lines look wrong as well for the same reason.

3. May 18, 2014

### Nikitin

E(L) = the expected loss, so that

Expected Profit = Expected gains - Expected losses = E(W)-E(L).

Is that a wrong line of thinking?

4. May 18, 2014

### Nikitin

Pls help i got my exam tomorrow

5. May 18, 2014

### Staff: Mentor

See my previous post...

6. May 18, 2014

### Nikitin

It's not the expected bet, but the expected loss. Each line is the probability for a certain outcome multiplied by the amount of money that would be lost in said outcome.

So in the first line, Lars wins on the first try and loses nothing. In the second line, Lars loses on the first try but wins on the second, losing one dollar. and so on until Lars loses 5 times in a row and gives up.

However, I get the wrong expected loss of money with this approach.

7. May 18, 2014

### Staff: Mentor

Yeah, but he still has to place his bet of 1 dollar to play the first round.
He uses 1+2 dollars (and wins 4).

8. May 18, 2014

### Nikitin

Ah yes, of course. He doesn't get his starting money back even if he wins (he just gets twice the original amount). That's where I was stuck. OK, i get the correct answer now. thanks