Find

the expected value and
the standard deviation in the following raffle:

Value Tickets

Grand Prize: $2,000.00 1

Second Place Prize: $800.00 20

Third Place Prize: $40.00 60

Fourth Place Prize: $30.00 300


Total Number of Tickets Sold: 10000

500

To calculate the expected value and standard deviation in this raffle, we need to assign a probability to each outcome and multiply it by the corresponding value.

First, let's calculate the probabilities. The probability of winning each prize is determined by the number of tickets sold for that prize divided by the total number of tickets sold.

For the Grand Prize:
Probability = 1/10,000

For the Second Place Prize:
Probability = 20/10,000 = 1/500

For the Third Place Prize:
Probability = 60/10,000 = 3/500

For the Fourth Place Prize:
Probability = 300/10,000 = 3/1000

Now, let's calculate the expected value. The expected value is the sum of the products of each outcome value and its corresponding probability.

Expected Value = (Value of Grand Prize * Probability of Grand Prize) + (Value of Second Place Prize * Probability of Second Place Prize) + (Value of Third Place Prize * Probability of Third Place Prize) + (Value of Fourth Place Prize * Probability of Fourth Place Prize)

Expected Value = ($2,000.00 * 1/10,000) + ($800.00 * 1/500) + ($40.00 * 3/500) + ($30.00 * 3/1000)

Expected Value = $0.20 + $1.60 + $0.24 + $0.09

Expected Value = $2.13

Therefore, the expected value of this raffle is $2.13.

Now, let's calculate the standard deviation. The standard deviation measures the dispersion or spread of the probability distribution.

Standard Deviation = sqrt((Value of Grand Prize - Expected Value)^2 * Probability of Grand Prize + (Value of Second Place Prize - Expected Value)^2 * Probability of Second Place Prize + (Value of Third Place Prize - Expected Value)^2 * Probability of Third Place Prize + (Value of Fourth Place Prize - Expected Value)^2 * Probability of Fourth Place Prize)

Standard Deviation = sqrt(($2,000.00 - $2.13)^2 * 1/10,000 + ($800.00 - $2.13)^2 * 1/500 + ($40.00 - $2.13)^2 * 3/500 + ($30.00 - $2.13)^2 * 3/1000)

Standard Deviation = sqrt(($1,997.87)^2 * 1/10,000 + ($797.87)^2 * 1/500 + ($37.87)^2 * 3/500 + ($27.87)^2 * 3/1000)

Standard Deviation = sqrt(3.99188369 + 3.19270861 + 0.31628951 + 0.02507192)

Standard Deviation = sqrt(7.52695373)

Standard Deviation = 2.744

Therefore, the standard deviation of this raffle is approximately 2.744.