800 hundred chances are sold at $2 apiece for a raffle. There is a grand prize of $650 dollars, two second prizes of $250, and five third prizes of $50. First calculate the expected value of the lottery. Determine if the lottery is a fair game. If the game is not fair, determine a price for playing the game that would make it fair.
several important questions affect the calculations
1. Are the winning tickets returned ? (more than likely they are not)
2. What is the order of drawing the winning tickets, that is, is the first price drawn last ? etc
To calculate the expected value of the lottery, we need to multiply the value of each outcome by its corresponding probability, and then sum those products.
First, let's determine the total amount of money collected from selling the chances. Since 800 chances are sold at $2 apiece, the total amount collected is 800 * $2 = $1600.
Next, let's determine the probability of winning each prize:
- Grand prize ($650) has a probability of 1/800, as there is only one grand prize out of 800 chances.
- Second prize ($250) has a probability of 2/800, as there are two second prizes out of 800 chances.
- Third prize ($50) has a probability of 5/800, as there are five third prizes out of 800 chances.
Now we can calculate the expected value:
Expected value = (Probability of grand prize * Value of grand prize) + (Probability of second prize * Value of second prize) + (Probability of third prize * Value of third prize)
Expected value = (1/800 * $650) + (2/800 * $250) + (5/800 * $50)
Expected value = $0.8125 + $0.625 + $0.3125
Expected value = $1.75
The expected value of the lottery is $1.75.
To determine if the lottery is a fair game, we compare the expected value with the cost of playing the game. In this case, the cost of playing the game is $2.
Since the expected value ($1.75) is less than the cost of playing the game ($2), the lottery is not a fair game. It means that, on average, a player would lose money by playing the lottery.
To make the game fair, we need to adjust the price of playing. The adjusted price should make the expected value equal to the cost of playing ($2).
Let's calculate the new price:
Expected value = Cost of playing the game
$1.75 = New price
Therefore, to make the lottery a fair game, the price for playing the game should be set to $1.75.