A box contains 3, red, 3 purple, 5 green, and 7 blue marbles. 2 marbles are sele

Question

A box contains 3, red, 3 purple, 5 green, and 7 blue marbles. 2 marbles are selected from the box without replacement. If both marbles are red or both marbles are pruple, you win $5. If both marbles are green, you win $4. If both marbles are blue, you win $2. If the marbles do not match in color, you lose $1. What is the expectation for this game?

I am stuck trying to figure out the probability for the marbles not matching in color.

I am not using combinatorics. I have all of the other marbles in a form similar to: amount of winnings ($) x (n/18 x ((n-1)/17)

Explanation / Answer

Probability of both red: (3/18)*(2/17) = 6/306

Probability of both purple: (3/18)*(2/17) = 6/306

Probability of both green: (5/18)*(4/17) = 20/306

Probability of both red: (7/18)*(6/17) = 42/306

Overall probablity of getting both same colors: 74/306

=> Probabilty of not getting same color: 1 - 74/306 = 232/306

Expectation : 5*(6/306) + 5*(6/306) + 4*(20/306) + 2*(42/306) + (-1)*(232/306) = -8/306 = - ($0.02614)

Expectation of the game is that you would lose $ 0.02614.

The other of calculating the probability is:

If we get red as first marble, then we should get any other color in second draw: (3/18)*((17-2)/17) = 45/306

If we get purple as first marble, then we should get any other color in second draw: (3/18)*((17-2)/17) = 45/306

If we get green as first marble, then we should get any other color in second draw: (5/18)*((17-4)/17) = 65/306

If we get blue as first marble, then we should get any other color in second draw: (7/18)*((17-6)/17) = 77/306

Adding the above probablities, we get 232/306. [same what we got above]

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