An urn contains 4 green, 6 blue and 10 yellow chips.
You pay $5 to draw a chip from the urn. Here are the rules of the game:
if you draw a green chip, the dealer returns you your bet and gives you an additional $5.
if you draw a blue chip, the dealer returns you your bet and gives you $1.
if you draw a yellow chip, the dealer keeps your $5.
What is the expected value of this game? (use formula for expected value of discrete distribution)
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To find the expected value of the game, we need to calculate the value of each outcome and multiply it by its corresponding probability.
Let's begin by calculating the probabilities of each outcome:
- Green chip: There are a total of 4 green chips out of 20 in the urn, so the probability of drawing a green chip is 4/20, which simplifies to 1/5.
- Blue chip: There are a total of 6 blue chips, giving us a probability of drawing a blue chip of 6/20, or 3/10.
- Yellow chip: With 10 yellow chips in the urn, the probability of drawing a yellow chip is 10/20, which simplifies to 1/2.
Now, let's calculate the value of each outcome:
- Green chip: Drawing a green chip gives you your bet ($5) back and an additional $5, resulting in a total gain of $10.
- Blue chip: Drawing a blue chip gives you your bet ($5) back and an additional $1, resulting in a total gain of $6.
- Yellow chip: Drawing a yellow chip means you lose your bet ($5), resulting in a total loss of $5.
Finally, we can calculate the expected value using the following formula:
Expected Value = (Value of Green Chip x Probability of Green Chip) + (Value of Blue Chip x Probability of Blue Chip) + (Value of Yellow Chip x Probability of Yellow Chip)
Expected Value = ($10 x 1/5) + ($6 x 3/10) + (-$5 x 1/2)
Expected Value = $2 + $1.8 - $2.5
Expected Value = $1.3
Therefore, the expected value of the game is $1.3.