In a box there are four prizes that are worth $30, three prizes worth $300, and one prize worth $1000.
A player will reach into the box and draw one of the prizes at random.
What is the fair price for this game?
To determine the fair price for the game, we need to calculate the expected value. The expected value of a random variable is the average value it would take on over a large number of repetitions of the experiment.
In this case, we have 4 prizes worth $30, 3 prizes worth $300, and 1 prize worth $1000. The probability of drawing each prize depends on the total number of prizes.
Let's calculate the expected value step by step:
1. Calculate the probability of drawing each prize:
- The probability of drawing a $30 prize = number of $30 prizes / total number of prizes = 4/8 = 1/2.
- The probability of drawing a $300 prize = number of $300 prizes / total number of prizes = 3/8.
- The probability of drawing a $1000 prize = number of $1000 prizes / total number of prizes = 1/8.
2. Calculate the expected value:
- Expected value = (probability of drawing a $30 prize * value of $30 prize) + (probability of drawing a $300 prize * value of $300 prize) + (probability of drawing a $1000 prize * value of $1000 prize)
- Expected value = (1/2 * $30) + (3/8 * $300) + (1/8 * $1000)
- Expected value = $15 + $112.50 + $125
- Expected value = $252.50
Therefore, the fair price for this game would be $252.50.