Question

A person participates in a state lottery by selecting six numbers from 1 through 51. If the six numbers match the six drawn by the lottery, regardless of order, then the participant wins the first prize of millions of dollars. If a participant's numbers match five of the six drawn, the participant wins second prize, which is not millions of dollars.
(a) Find the probability of winning first prize. (Round your answer to ten decimal places.)
(b) Find the probability of winning second prize. (Round your answer to seven decimal places.)

Answer 1

There are

$\dbinom{51}6=\dfrac{51!}{6!(51-6)!}=18,009,460$

total ways to draw any 6 numbers from the range 1 to 51, regardless of order.

Given 6 selected numbers that match those drawn by the lottery, there are

$6!=720$

ways of rearranging them. So the probability of winning 1st prize is

$\dfrac{720}{18,009,460}=\dfrac{36}{900,473}\approx0.0000399790$

Next, given 6 selected numbers of which 5 match those drawn by the lottery, there are

$5!=120$

ways of rearranging those 5 matching numbers. There are 46 remaining numbers that didn't get drawn, so the probability of winning 2nd prize is

$\dfrac{46\cdot120}{18,009,460}=\dfrac{12}{39,151}\approx0.0003065$