Question

Suppose that you enter a raffle drawing where is costs $5 to enter. There are 1500 tickets sold in all and there are fifteen $50 prizes, ten $100 prizes, and two $500 prizes. In contrast to previous example, there is only one raffle drawing and you purchase one ticket. Let x be the discrete random variable that represents the possible winnings of x = $0, x = $50, x = $100, and x = $500.

1. Create a table for the probability distribution of x.

2. Use this table to calculate the expected value.

3. Write an interpretation of this value.

4. If we include the initial entry fee, what does this say about the quality of this type of “investment?”

e. Suppose that the top prize of $500 was to be changed while leaving the other prize amounts unchanged. If we ignore the entry fee, to what value should this top prize be changed to make the expected value be exactly $5? Show your work.

Answer 1

Probability = Favourable Outcomes/Total Outcomes

There are 1500 tickets sold in Total.

Out of which, 15 will win $50 , 10 will win $100 and 2 will win $500 and the rest 1473 will win $0.

Thus, Probability Distribution of x becomes:

X	P(x)
$0	0.982
$50	0.01
$100	0.007
$500	0.001
Total	1

Expected Value of X = X*p(x)

X	P(x)	X*P(x)
0	0.982	0
50	0.010	0.5
100	0.007	0.7
500	0.001	0.5
Total	1	1.7

Thus, Expected value of X is $1.7.

Interpretation: In the long run, over an average of $1.7 can be won by investing.

When we include initial entry fees ($5) in this investment, the Expected value of x will reduced by $5 overall and thus it will result in $ 3.3(1.7-5) loss by investing here.

e. Now Suppose $500 changed to $a such that Expected value of x is $5 ignoring the entry fees.

X	P(x)	X*P(x)
0	0.982	0
50	0.010	0.5
100	0.007	0.7
a	0.001	0.00a
Total	1	1.2 + 0.00a

Expected value of x = X*P(x)

5 = 1.2 + 0.00a

3.8 = 0.00a

3800 = a

Thus, Top prize (I.e.$500) should be changed to $3800 so that Expected value becomes $5 after ignoring initial entry fees.

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