A state runs a lottery in which six numbers are randomly selected from 40 without replacement. A player chooses six numbers before the state’s sample is selected.

(a) What is the probability that the six numbers chosen by a player match all six numbers in the state’s sample?

(b) What is the probability that five of the six numbers chosen by a player appear in the state’s sample?

(c) What is the probability that four of the six numbers chosen by a player appear in the state’s sample?

(d) If a player enters one lottery each week, what is the expected number of weeks until a player matches all six numbers in the state’s sample?

Step 1 of 5:

It is given that a state runs a lottery and 6 numbers are randomly selected from 40 numbers without replacement.

Also,it is given that a player chooses 6 numbers before the state selects the sample.

Using this we need to find the required values.

Step 2 of 5:

(a)

Here we have to find the probability that all the six numbers chosen by the player will match all the numbers selected by the state.

Let X be the numbers chosen .Then X follows the HyperGeometric distribution with parameters N=40,n=6,K=6.

The probability mass function of the HyperGeometric distribution is

P(X=x)=

Now we have to find the value of P(X=6).

P(X=6)=

=

=

=

=0.0000002605

Thus, the probability that all the 6 numbers chosen by the player matches the numbers chosen by the state is 0.0000002605.

Step 3 of 5:

(b)

Here we have to find the probability that 5 out of 6 numbers chosen by the player matches the numbers chosen by the state.

That is we have to find P(X=5) and it is given by

P(X=5)=

=

=

=

=

=0.00005315

Thus, the probability that 5 of the 6 numbers chosen by the player matches the number chosen by the state is 0.00005315.