Question

In a certain lottery, six numbers are randomly chosen form the set {0, 1, 2, ..., 49} (without replacement). To win the lottery, a player must guess correctly all six numbers but it is not necessary to specify in which order the numbers are selected.

(a) What is the probability of winning the lottery with only one ticket?

(b) Suppose in a given week, 6 million lottery tickets are sold. Suppose further that each player is equally likely to choose any of the possible number combinations and does so independent of the selections of all other players. What is the probability that exactly four players correctly select the winning combination?

(c) Again assuming 6 million tickets sold, what is the most probable number of winning tickets?

(d) Repeat parts (b) and (c) using the Poisson approximation to the binomial probability distribution. Is the Poisson distribution an accurate approximation?

Answer 1

(a)

Probability of winning the lottery with one ticket is given by

(b)

Suppose, random variable X denotes number of players who correctly selected the winning combination.

We define correctly selecting the winning combination as success.

Selecting the combination by a player is independent of other players.

Probability that exactly four players correctly selected the winning combination is given by

[Using R-code 'dbinom(4,6000000,0.00000007151124)']

(c)

We know, for Binomial probability distribution, mode is given by .

So, most probable number of winning tickets

Hence, most probable number of winning tickets is 0.

(d)

Here, n = 6000000 is very large and ,p = 0.00000007151124 is very small. But np = 0.4290674 is finite.

So, Poisson approximation is accurate.

So, probability that exactly four players correctly selected the winning combination is given by

[Using R-code 'dpois(4,0.4290674)']

We know, for Poisson probability distribution, mode is given by .

So, most probable number of winning tickets

Hence, most probable number of winning tickets is 0.