COLLEGE MATH PLEASE HELP

Question

A bag contains 5 marbles: 2 blue and 3 red. A player pays $5 to draw from the bag twice without replacement. If the player draws one or more blue marbles he wins $7.

Give answers rounded to the nearest hundredth (two decimal places).  Examples: If the answer is 7, type 7.00. If the answer is negative 45 cents type -0.45. 

a. Find the expected return to the player. $ [x]                .

b. Find the fair price of playing the game. $  [y]    

Answer 1

The first result means we drew a blue on our first draw, which was 2 out of 5 marbles, so the chance of us picking blue on our first draw is 2/5 or 0.4. This wins us the $7 already, so we don't care about the second draw.

The second result means we drew a red on our first draw, which was 3 out of 5 marbles, so the chance of us picking red on the first draw is 3/5 or 0.6. This does not win us our money yet, so we will have to draw the second time.

Stemming from the second result we have:

The result 2A means that on our second draw we drew a blue, which was 2 out of 4 marbles, so the chance of us picking blue on the second draw is 1/2 or 0.5. However, the overall chance of this whole situation happening (picking red, then blue), is the chance of drawing red on the first (0.6) times the chance of drawing blue on the second (0.5), which turns out to be 0.3. This wins us our $7.

The result 2B means that on our second draw we drew a red, which was 2 out of 4 marbles, so the chance of us picking blue on the second draw is 1/2 or 0.5. However, the overall chance of this whole situation happening (picking red, then red again), is the chance of drawing red on the first (0.6) times the chance of drawing red on the second (0.5), which turns out to be 0.3. This does not win us our money and the game is over.

Thus the chance we will win is the sum of all the possibilities by which we could win: Drawing blue on the first try (0.4) and drawing blue on the second try (0.3), which equates to 0.7 or 7/10.

Now to determine the expected return, we think about the money. We start the game 5 dollars down. Then there is a 0.7 chance we earn $7, so we have multiply 0.7 by $7 to get the expected amount to money gained, which is $4.90. So it is expected (we might say "on average") that someone earns $4.90 but loses $5, which is in total $4.90 - $5 which makes -$0.10, or a ten cent loss. That is the answer to part a.

The fair price would be the price equal to the expected amount of money to be gained, $4.90, which is the answer to part b.

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