Question

Jack and Jill gamble on a roll of a die, as follows. If the die shows 1 or 2 spots, Jack gives Jill $1. If the die shows 5 or 6 spots, Jill gives Jack $1. If the die shows 3 or 4 spots, no money changes hands. Suppose Jack and Jill play this game 400 times. The chance that Jill’s net gain is more than $20 is closest to:
10%, 20%, 30% or 40%?

Answer 1

Look at it this way:

A player is rolling dice.
Every time the player rolls a 1 or 2, the player gets a dollar.
Every time the player rolls a 5 or 6, the player loses a dollar.
Every time the player rolls a 3 or 4, there is no change.

Over time, I think there's very little chance of the player making any significant gain.

Answer 2

Thank you - please put it down to incredible stupidity - I just wasn't thinking!

Answer 3

In short, 10% chance of getting $20

Answer 4

what do think?

Answer 5

Right answer is 10%

Answer 6

The answer is correct at 10%, but the thinking is wrong. First consider the expected result. That is 0 (i.e. 1/3 @ -1, 1/3 @ 0 and 1/3 @ 1). Then figure out the Standard Error. In this case, it is \[\sqrt{\frac{ 2 }{ 3 }}\] which equals 0.8165. Then you multiply this by the square root of the number of draws (400) which gives a SD of 16.3299. So, the expected winnings is 0, with a SD of 16.3299. Then you can figure out the Z score as follows:
\[z = \frac{ 20-0 }{ 16.3299 } = 1.2247\]
Once you have the z score, you can look at the area under the normal curve and you get a result of 11.03%