Prizes and the chances of winning in a sweepstakes are given

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

Solution

Answer.

Expected Value = P (probability) . Value

Random variable X with values x₁, x₂, x₃, . . . x_n, and
respective Probabilities of p₁,p₂, p₃, . . . p_n,

the expected value of X is given by the formula:
E(X) = x₁p₁ + x₂p₂ + x₃p₃ + . . . + x_np_n.

Now for

Ans. a

x_{1 =} $ 20,000,000 with p₁  1/300,000,000.
x_{2 =} $ 450,000 with p₂  1/200,000,000.
x_{3 =} $ 50,000 with p₃ 1/20,000,000.
x_{4 =} $ 5,000 with p₄  1/4,000,000.
x_{5 =} $ 800 with p₅ 1/700,000.
x_{6 =} $ 90 with p₆ 1/3,000.

Now, E(X) = x₁p₁ + x₂p₂ + x₃p₃ + x₄p₄ + x₅p₅ + x₆p₆
= $ 20,000,000 . 1/300,000,000 +$ 450,000 . 1/200,000,000 + $ 50,000 . 1/20,000,000 +
$ 5,000 .1/4,000,000 + $ 800 .  1/700,000 + $ 90 .1/3,000.

= 2/30 + 45/20,000 + 5/2,000 + 5/4,000 + 8/7000 + 9/3000
= 0.067 + 0.002 + 0.002 + 0.001 + 0.001 +0.003
E(X) = 0.076

Ans. b

Now these are the 6 entries which are Awarded but other Entries cost Loss of 34 cents.
So overall E(X) will be,
E(X) obatined above for the 6 entries + E(X) of other entries

E(X) of other entries will be,
300,000,000 - 6 = 299,999,994 values
and each with Probability of 1 - (p₁ + p₂ + p₃ + p₄ + p₅ + p₆)

(p₁ + p₂ + p₃ + p₄ + p₅ + p₆) will be,
= 1/1000 * (1/300,000 + 1/200,000 + 1/20,000 + 1/4,000 + 1/700 + 3 )
= 1/1000 * (0.000003 + 0.000005 + 0.00005 + 0.00025 + 0.0014 + 3 )
= 0.0003001708
1 - 0.0003001708 = 0.9996998292 (approx. = 0.999)

E(X) of other entries,
= 299,999,994 * (-34 cents . 0.999)
= 299,999,994 * (-33.967)
= 10190099796.198

Finally,
E(X) obatined above for the 6 entries + E(X) of other entries
= 0.076 - 10190099796.198
= 10190099796.122