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Math  /  Data & Statistics

QuestionAngelo's Pizza is having a ticket raffle to raise money for new uniforms for the soccer team they sponsor. Each ticket gives an equal chance to win one $25\$ 25 gift certificate, one $15\$ 15 gift cercificatte, or one of three $5\$ 5 gift certificates. Leon buys one $1\$ 1 ticket. What is Leon's expected profit or loss if 100 total tickets are sold? a loss of $0.55\$ 0.55 a profit of $0.55\$ 0.55 a profit of $0.45\$ 0.45 a loss of $0.45\$ 0.45

Studdy Solution

STEP 1

1. There are a total of 100 tickets sold.
2. Each ticket has an equal chance of winning any of the prizes.
3. The prizes are one $25\$ 25 gift certificate, one $15\$ 15 gift certificate, and three $5\$ 5 gift certificates.
4. Leon buys one ticket for $1\$ 1.
5. We need to calculate the expected profit or loss for Leon.

STEP 2

1. Calculate the probabilities of winning each prize.
2. Calculate the expected value of the prize.
3. Calculate Leon's expected profit or loss by subtracting the ticket cost from the expected value of the prize.

STEP 3

Calculate the probability of winning the $25\$ 25 gift certificate. Since there is only one $25\$ 25 gift certificate and 100 tickets: P($ 25 prize)=1100 P(\text{\$ 25 prize}) = \frac{1}{100}

STEP 4

Calculate the probability of winning the $15\$ 15 gift certificate. Since there is only one $15\$ 15 gift certificate and 100 tickets: P($ 15 prize)=1100 P(\text{\$ 15 prize}) = \frac{1}{100}

STEP 5

Calculate the probability of winning one of the three $5\$ 5 gift certificates. Since there are three $5\$ 5 gift certificates and 100 tickets: P($ 5 prize)=3100 P(\text{\$ 5 prize}) = \frac{3}{100}

STEP 6

Calculate the probability of winning no prize. Since there are 100 tickets and only 5 prizes, the probability of not winning any prize is: P(no prize)=1−(1100+1100+3100)=1−5100=95100 P(\text{no prize}) = 1 - \left( \frac{1}{100} + \frac{1}{100} + \frac{3}{100} \right) = 1 - \frac{5}{100} = \frac{95}{100}

STEP 7

Calculate the expected value of the prize. The expected value E(X) E(X) is the sum of each prize amount multiplied by its probability: E(X)=25⋅1100+15⋅1100+5⋅3100+0⋅95100 E(X) = 25 \cdot \frac{1}{100} + 15 \cdot \frac{1}{100} + 5 \cdot \frac{3}{100} + 0 \cdot \frac{95}{100}

STEP 8

Simplify the expression for the expected value: E(X)=25100+15100+15100=55100=0.55 E(X) = \frac{25}{100} + \frac{15}{100} + \frac{15}{100} = \frac{55}{100} = 0.55

STEP 9

Calculate Leon's expected profit or loss by subtracting the ticket cost from the expected value of the prize: Expected profit/loss=E(X)−1=0.55−1=−0.45 \text{Expected profit/loss} = E(X) - 1 = 0.55 - 1 = -0.45
The expected loss for Leon is $0.45\$ 0.45.

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