Question

prizes and the chances of winning in a sweepstakes are given in the table below.

(a) find the expected value (in dollars) of the amount won by one entry

(b) find the expected value (in dollars) if the cost of entering this sweepstakes is the cost of a postage stamp (34 cents)

Explanation:

Step1: Calculate the probability of each prize

• For \( \$10,000,000 \): \( P_1=\frac{1}{400000000} \)
• For \( \$150,000 \): \( P_2 = \frac{1}{100000000} \)
• For \( \$25,000 \): \( P_3=\frac{1}{80000000} \)
• For \( \$15,000 \): \( P_4=\frac{1}{4000000} \)
• For \( \$800 \): \( P_5=\frac{1}{500000} \)
• For \( \$45 \): \( P_6=\frac{1}{9000} \)

Step2: Calculate the expected value of the amount won (part a)

The formula for expected value \( E(X)=\sum_{i = 1}^{n}x_iP_i \)

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Step3: Calculate the expected value considering cost (part b)

The cost \( C = 0.34 \) dollars. The new expected value \( E'(X)=E(X)-C \)

$$ E'(X)=0.0371625 - 0.34=- 0.3028375 $$

Answer:

a. \( 0.0371625 \)
b. \( - 0.3028375 \)