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 c

Explanation:

Step1: Calculate the probability for each prize

The probability \(P\) of an event is given by \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
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 a \(\$45\) - watch: \(P_6=\frac{1}{9000}\)

Step2: Use the expected - value formula \(E(X)=\sum_{i = 1}^{n}x_iP_i\)

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Find a common denominator (the least common multiple of \(40,2000,3200,800,2500,200\) is \(160000\))

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Step3: For part (b)

If the cost of entering is \(C = 0.34\) dollars, the new expected value \(E'(X)=E(X)-C\)
\(E'(X)=0.03409125- 0.34=-0.30590875\)

Answer:

(a) The expected value of the amount won by one entry is \(\$0.03409125\)
(b) The new expected value is \(-\$0.30590875\)