Assume that the pond contains 120 fish: 97 green, 22 yellow, and 1 red. A contestant pays $0.60 to randomly catch one fish and receives $0.40 if the fish is green, $0.95 if the fish is yellow, and $11.00 if the fish is red.
How much (on average) does the carnival gain on each play?
In the long run, it would be equivalent to 120 plays with
120*0.60=72 as revenue and the following payouts:
97 green at 0.95
22 yellow at 0.40
1 red at 11.00
Subtract payouts from revenue and divide by 120 plays to get gain on individual play.
Well, let's calculate the average gain for the carnival.
The probability of catching a green fish is 97/120, and the carnival gains $0.60 - $0.40 = $0.20 for each green fish caught. So, the average gain for each green fish caught is (97/120) * ($0.20) = $0.1617.
The probability of catching a yellow fish is 22/120, and the carnival gains $0.60 - $0.95 = -$0.35 (negative value) for each yellow fish caught. So, the average gain for each yellow fish caught is (22/120) * (-$0.35) = -$0.0642.
Finally, the probability of catching a red fish is 1/120, and the carnival gains $0.60 - $11.00 = -$10.40 (again, negative value) for each red fish caught. So, the average gain for each red fish caught is (1/120) * (-$10.40) = -$0.0867.
Now, let's calculate the overall average gain by summing up the gains for each type of fish: $0.1617 + (-$0.0642) + (-$0.0867) = $0.0108.
So, on average, the carnival gains $0.0108 for each play. Hooray for the carnival!
To calculate the average gain for each play, we need to find the expected value for the amount of money the carnival will gain.
First, let's find the probability of catching each color fish.
The probability of catching a green fish = number of green fish / total number of fish = 97 / 120 = 0.8083 or 80.83%
The probability of catching a yellow fish = number of yellow fish / total number of fish = 22 / 120 = 0.1833 or 18.33%
The probability of catching a red fish = number of red fish / total number of fish = 1 / 120 = 0.0083 or 0.83%
Next, let's calculate the expected value for each outcome.
Expected value for the green fish = probability * amount of money = 0.8083 * $0.40 = $0.3233
Expected value for the yellow fish = probability * amount of money = 0.1833 * $0.95 = $0.1745
Expected value for the red fish = probability * amount of money = 0.0083 * $11.00 = $0.0913
Now, let's calculate the overall expected value for each play by summing up the expected values for each outcome.
Overall expected value = (expected value for green fish) + (expected value for yellow fish) + (expected value for red fish)
= $0.3233 + $0.1745 + $0.0913
= $0.5891
Therefore, on average, the carnival gains $0.5891 on each play.
To find out how much the carnival gains on average for each play, we need to calculate the expected value. The expected value is the sum of each outcome multiplied by its respective probability.
Let's start by calculating the probabilities of catching each type of fish:
Probability of catching a green fish = Number of green fish / Total number of fish
Probability of catching a yellow fish = Number of yellow fish / Total number of fish
Probability of catching a red fish = Number of red fish / Total number of fish
Probability of catching a green fish = 97 / 120 = 81%
Probability of catching a yellow fish = 22 / 120 = 18.33%
Probability of catching a red fish = 1 / 120 = 0.83%
Now, let's calculate the expected value for each outcome:
Expected value of catching a green fish = Probability of catching a green fish * Payout for catching a green fish
Expected value of catching a yellow fish = Probability of catching a yellow fish * Payout for catching a yellow fish
Expected value of catching a red fish = Probability of catching a red fish * Payout for catching a red fish
Expected value of catching a green fish = 0.81 * $0.40 = $0.324
Expected value of catching a yellow fish = 0.1833 * $0.95 = $0.1738
Expected value of catching a red fish = 0.0083 * $11.00 = $0.0913
Now, let's find the overall expected value by summing up the expected values of each outcome:
Overall expected value = Expected value of catching a green fish + Expected value of catching a yellow fish + Expected value of catching a red fish
Overall expected value = $0.324 + $0.1738 + $0.0913 = $0.5891
Therefore, the carnival gains an average of $0.5891 for each play.