"Marble Madness" is a local carnival game, costing $2. There are 100 total marbles in a bag: 2 red, 8 orange, 10 yellow, 30 green, 30 blue, and 20 black. If a red marble is pulled, you win $6, an orange marble wins $4, and a yellow marble wins $2. A green, blue, and black marble result in a loss. What is the expected value of a marble pull?
To calculate the expected value, we need to calculate the probability of each outcome and multiply it by the corresponding payoff, then sum them all up.
The probability of pulling a red marble is 2/100, and the payoff is $6. So the contribution to the expected value from the red marble is (2/100) * $6 = $0.12.
The probability of pulling an orange marble is 8/100, and the payoff is $4. So the contribution to the expected value from the orange marble is (8/100) * $4 = $0.32.
The probability of pulling a yellow marble is 10/100, and the payoff is $2. So the contribution to the expected value from the yellow marble is (10/100) * $2 = $0.20.
The probability of pulling a green marble is 30/100, and the payoff is -$2 (loss). So the contribution to the expected value from the green marble is (30/100) * -$2 = -$0.60.
The probability of pulling a blue marble is 30/100, and the payoff is -$2 (loss). So the contribution to the expected value from the blue marble is (30/100) * -$2 = -$0.60.
The probability of pulling a black marble is 20/100, and the payoff is -$2 (loss). So the contribution to the expected value from the black marble is (20/100) * -$2 = -$0.40.
Now, let's sum up all the contributions to calculate the expected value:
Expected value = $0.12 + $0.32 + $0.20 - $0.60 - $0.60 - $0.40
Expected value = $-0.96
So the expected value of pulling a marble from the bag is -$0.96. This means that, on average, each marble pull is expected to result in a loss of $0.96.
To find the expected value of a marble pull, we need to calculate the probability of pulling each color marble and then multiply it by the corresponding amount won or lost.
Step 1: Calculate the probability of pulling each color marble:
- Red marble: 2 out of 100 marbles = 2/100 = 1/50
- Orange marble: 8 out of 100 marbles = 8/100 = 2/25
- Yellow marble: 10 out of 100 marbles = 10/100 = 1/10
- Green marble: 30 out of 100 marbles = 30/100 = 3/10
- Blue marble: 30 out of 100 marbles = 30/100 = 3/10
- Black marble: 20 out of 100 marbles = 20/100 = 1/5
Step 2: Calculate the expected value for each color marble:
- Red marble: $6 * (1/50) = $0.12
- Orange marble: $4 * (2/25) = $0.32
- Yellow marble: $2 * (1/10) = $0.20
- Green marble: -$2 * (3/10) = -$0.60
- Blue marble: -$2 * (3/10) = -$0.60
- Black marble: -$2 * (1/5) = -$0.40
Step 3: Calculate the overall expected value:
- Summing the expected values for each color marble:
$0.12 + $0.32 + $0.20 - $0.60 - $0.60 - $0.40 = -$1.96
The expected value of pulling a marble from the bag is -$1.96. This means that, on average, you can expect to lose $1.96 per marble pull.