Although many people buy lottery tickets out of an expectati

Although many people buy lottery tickets out of an expectation
of good luck, probabilistically speaking, buying lottery tickets is usu-
ally a waste of money. Indeed, suppose that in a weekly state lottery
ve of the numbers 0, 1, ..., 49 are selected without replacement at ran-
dom and someone who holds exactly those numbers wins the lottery.
Please compute the following (no need to do arithmetic calculations).
(a) (5pt) If you hold one ticket, what is the probability that you\'re a
winner in a given week?
(b) (5pt) Now, assume that weekly lotteries are independent and that
a year consists of 52 weeks. If you keep buying a ticket every week
for 40 years, what is the probability that you\'re going to win a
lottery in at least one week?

Solution

Suppose that in a weekly state lottery five of the numbers 0 , 1 , .....49 are selected without replacement at random. Someone who holds exactly those numbers wins the lottery.

P[ someone holding one ticket will be the winner] =   1/C(50 ,5 )

Since, 0, 1 , 2 ................49 are total numbers 50

Here I have used combination method to select r objects from n

i.e.   n = 50    and   r = 5

b) Suppose this person buys a ticket every week for 40 years.

Then, P[ He will win the lottery atleast one week] = 1 – [1 – (1/C(50,5)]^(52*40)

                                                                                          = 0.00098