Question

A lottery with 200 tickets sold gives five prizes-grand prize of $50 2 second prizes of $20 each and 2/3 prices of $5 each. Tickets for the prices are drawn at random, without replacement (no one can win two prizes). Define a random variable X as the amount won by a person who holds one ticket.

A)give the probability distribution for x

B)if a ticket cost $10 what is the probability of making a profit by buying a ticket?

Answer 1

A)

The probability to win $50 prize = 1/200

The probability to win $20 prize = 2/200 = 1/100

The probability to win $5 prize = 2/200 = 1/100

Probability of not winning any prize = 1 - (1/200 + 1/100 + 1/100) = 39/40

The probability distribution for x is,

P(X = 50) = 1/200

P(X = 20) = P(X = 5) = 1/100

P(X = 0) = 39/40

B)

Expected value of X is,

E[X] = 50 * 1/200 + 20 * 1/100 + 5 * 1/100 + 0 * 39/40 = 0.5

E[X²] = 50² * 1/200 + 20² * 1/100 + 5² * 1/100 + 0² * 39/40 = 16.75

Variance of X is,

Var[X] = E[X²] - E[X]²  = 16.75 - 0.5² = 16.5

Standard deviation of X is,

SD[X] = = 4.062

Assuming normal distribution, probability of making a profit by buying a ticket

= P(X > 10)

= P[Z > (10 - 0.5) / 4.062]

= P[Z > 2.34]

= 0.0096