In a carnival game, the player selects two coins from a bag containing four silver dollars and twelve slugs.
(a) Write down the probability distribution for the winnings.
(b) Determine how much the player would have to pay so that he would break even, on the average, over many repetitions of the game.
To solve this problem, let's start with part (a):
(a) The player selects two coins from the bag. There are four silver dollars and twelve slugs in the bag. Since there are 16 coins in total, the probability of selecting a silver dollar on the first draw is 4/16 = 1/4, and the probability of selecting a slug is 12/16 = 3/4.
After the first draw, there will be 15 coins left in the bag, with the number of silver dollars and slugs decreasing accordingly. The probability of selecting a silver dollar on the second draw depends on the outcome of the first draw.
If the first coin is a silver dollar, there will be 3 silver dollars left out of the remaining 15 coins, so the probability of selecting a silver dollar on the second draw is 3/15 = 1/5. If the first coin is a slug, there will be still 4 silver dollars left out of the remaining 15 coins, so the probability of selecting a silver dollar on the second draw is 4/15.
Therefore, the probability distribution for the winnings is as follows:
- Selecting two silver dollars: (1/4) * (1/5) = 1/20
- Selecting one silver dollar and one slug: (1/4) * (4/15) = 4/60 = 1/15
- Selecting two slugs: (3/4) * (11/14) = 33/56
Now let's move on to part (b):
(b) To determine how much the player would have to pay to break even on average, we need to calculate the expected winnings.
First, let's assign values to the winnings. Let's say the player wins $2 for selecting two silver dollars, loses $1 for selecting one silver dollar and one slug, and loses $2 for selecting two slugs.
Using the probability distribution from part (a), we can calculate the expected winnings as follows:
Expected winnings = (probability of selecting two silver dollars) * (amount won for two silver dollars) + (probability of selecting one silver dollar and one slug) * (amount won for one silver dollar and one slug) + (probability of selecting two slugs) * (amount won for two slugs)
Expected winnings = (1/20) * $2 + (1/15) * (-$1) + (33/56) * (-$2)
Simplifying and calculating:
Expected winnings = $0.10 - $0.07 - $1.18 = -$1.15
The negative expected winnings indicate that, on average, the player would lose $1.15 per game.
To break even on average, the player would have to pay an amount equal to the negative value of the expected winnings, which is $1.15.
So, the player would have to pay $1.15 to break even, on average, over many repetitions of the game.