Prizes and the chances of winning in a sweepstakes are given in the table below.
ID: 3309445 • Letter: P
Question
Prizes and the chances of winning in a sweepstakes are given in the table below.
Prize
Chances
$20,000,000
1 chance in 200,000,000
$150,000
1 chance in 150,000,000
$75,000
1 chance in 70,000,000
$5,000
1 chance in 4,000,000
$800
1 chance in 700,000
A watch valued at $65
1 chance 8,000
(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)
Prize
Chances
$20,000,000
1 chance in 200,000,000
$150,000
1 chance in 150,000,000
$75,000
1 chance in 70,000,000
$5,000
1 chance in 4,000,000
$800
1 chance in 700,000
A watch valued at $65
1 chance 8,000
Explanation / Answer
(a) Find the expected value (in dollars) of the amount won by one entry:-
1 chance in 200,000,000 is basically another way of saying with probability = 1/200,000,000.
In order to find expected value we must simply multiply the value by the probability of winning that amount and sum them together:
Therefore E(X) = X_1*P(X_1)+X_2*P(X_2)+....
E(X) = ($20,000,000) * (1/200,000,000)
+ ($150,000) * (1/150,000,000)
+ ($75,000) * (1/70,000,000)
+ ($5,000) * (1/4,000,000)
+ ($800) * (1/700,000)
+ ($65) * (1/8,000)
= 1 + 0.001 + 0.0010 + 0.0012 + 0.0011 + 0.0081
E(X) = $ 1.0124
(b) Find the expected value (in dollars) if the cost of entering this sweepstakes is the cost of a postage stamp (34 cents):-
To find value of this we must substract 34 cents from the mean we got
But mean we got is in dollars
Therefore $1 = 100 cents
$1.0124 = 1.0124 * 100
= 101.24 cents
Therefore 101.24 - 34 cents
= 67.24 cents.
= $0.6724
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