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(1 point) Prizes and the chances of winning in a sweepstakes are given in the table below.
Prize Chances
$15,000,000 1 chance in 200,000,000
$250,000 1 chance in 100,000,000
$50,000 1 chance in 10,000,000
$20,000 1 chance in 4,000,000
$900 1 chance in 100,000
A watch valued at $60 1 chance in 9,000
(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)

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Edufirst

Answer:

  • Part a) Expected value of the amount won: $0.10

  • Part b) Expected value if the cost is 34cents: -$0.24 (loss).

Explanation:

Part a)

The expected value, E(x) is the sum of product of each outcome by its probability:

     [tex]E(X)=\sum\limits^n_{i=1} {x_i\cdot P(x_i)}[/tex]

$15,000,000 × 1 / 200,000,000 + $250,000 × 1 / 100,000,000 +

+ $50,000 × 1 / 10,000,000 + $20,000 × 1 / 4,000,000 +

+ $900 × 1 / 100,000 + $60  × 1 / 9,000 =

= $0.075 + $0.0025 + $0.005 + $0.005 + $0.009 + $0.00667 = $0.10

Part b)

You have to subtract the cost of the stamp: $0.34

  • $0.10 - $0.34 = - $0.24

Part(a):The expected value is 0.164

Part(b):The expected value is -0.176

Expected value:

The expected value is a long-run average value of random variables. It also indicates the probability-weighted average of all possible values.

Part(a):

The probability of not getting any prize is,

[tex]1-(\frac{1}{100,000,000}+\frac{1}{10,000,000}+\frac{1}{ 4,000,000}+\frac{1}{ 100,000}+ \frac{1}{ 9,000})=0.99988[/tex]

The expected value is,

[tex]\sum XP(X)=(15,000,000\times 0.000000005)+(250,000\times 0.0000001)+(50000\times 0.0000001)+(20000\times 0.0000025)+(900\times 0.00001)\\=0.164[/tex]

Part(b):

Cost of entering is 34 cents= $ 0.34

So, the expected value is,

[tex]0.164-0.34=-0.176[/tex]

Learn more about the topic expected value:

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