Question

We have a lottery with three prizes. Two of them are $30 and the grand is $200. The total number of tickets is 70 and one for each person. No one can win more than a prize. When the tickets are pulled, the first two prizes of $30 are awarded, then the grand prize.

a) how many ways we can award the prizes?

b) Suppose a person has one ticket, what's the probability of winning any prize?

c) If a person has 40 tickets, what's the probability of winning the grand prize?

d) suppose you can return each ticket you win. There's no limit to winning. If a person takes 25 tickets. what's the probability s/he wins a prize, but not the grand prize?

Answer 1

a) The number of ways that the prize can be awarded is computed here as:
= Total number of ways to select two $30 prizes out of 70 tickets * Number of ways to select the grand prize from remaining 68 tickets

$= \binom{70}{2}*68 = 164220$

b) The probability of winning any price here is computed as:

= Total number prizes / Total number of tickets

= 3/70

Therefore 3/70 = 0.0429 is the required probability here.

c) Probability of winning the grand prize with 40 tickets is computed here as:

= Total number of tickets which could have the grand prize / Total number of tickets

= 40/70

= 4/7

Therefore 4/7 = 0.5714 is the required probability here.

d) Given that the grand prize is not won, the probability that he/she wins a prize is computed here as:

= 1 - Probability that no prize is won from those 25 tickets given that there is no grand prize won on that ticket

= 1 - ( Total ways to select 2 tickets from remaining (69 - 25) tickets with no grand prize / Total ways to select 2 tickets from 69 tickets)

$= 1 - \frac{\binom{44}{2}}{\binom{69}{2}} = 0.5968$

Therefore 0.5968 is the required probability here.