Question

4. In the Texas All or Nothing, twelve numbers are selected (without repetition) from 1 through 24 and order does not matter. (a.) How many possible tickets are there? (b.) You win the grand prize if all 12 numbers selected match the 12 you picked or if none of the numbers selected match the ones you picked. What is the probability of winning the grand prize? (c.) You win $500 if you match exactly 11 numbers or exactly 1 number. What is the probability you win $500?

Answer 1

a) There are 24 possible numbers, from which we select 12. The numbers are no repeated and the order does not matter

The possible number of tickets are

$\binom{24}{12}=\frac{24!}{12!(24-12)!}=2704156$

ans: The possible number of tickets are 2704156

b) Number of winning numbers =12

Number of  not winning numbers = 24-12=12

The possible number of ways in which we can select a ticket with 12 winning numbers (and 0 not winning numbers) is

$\binom{12}{12}\times \binom{12}{0}=\frac{12!}{12!(12-12)!}\times \frac{12!}{0!(12-0)!}=1$

The possible number of ways in which we can select a ticket with 12 not winning numbers (and 0 winning numbers) is

$\binom{12}{0}\times \binom{12}{12}=\frac{12!}{0!(12-0)!}\times \frac{12!}{12!(12-12)!}=1$

Total number of ways of getting a grand prize = (number of ways of getting a ticket with 12 winning numbers) +(number of ways of getting a ticket with 12 non winning numbers) = 1+1 =2

The probability of winning the grand prize is

$\begin{align*} P(\text{win grand prize})&=\frac{\text{total number of ways of getting a grand prize}}{\text{number of possible tickets}}\\&=\frac{2}{2704156}\\ &= 0.00000074 \end{align*}$

ans:  The probability of winning the grand prize is 0.00000074

c) The possible number of ways in which we can select a ticket with 11 winning numbers and 1 not winning numbers (that is match exactly 11)  is

$\binom{12}{11}\times \binom{12}{1}=\frac{12!}{11!(12-11)!}\times \frac{12!}{1!(12-1)!}=144$

The possible number of ways in which we can select a ticket with 1 winning number and 11 not winning numbers (that is match exactly 1)  is

$\binom{12}{1}\times \binom{12}{11}=\frac{12!}{1!(12-1)!}\times \frac{12!}{11!(12-11)!}=144$

Total number of ways of winning $500 = (number of ways of getting a ticket with 11 winning numbers) +(number of ways of getting a ticket with 1 winning number) = 144+144 =288

The probability of winning $500 is

$\begin{align*} P(\text{win \$500})&=\frac{\text{total number of ways of winning \$500}}{\text{number of possible tickets}}\\&=\frac{288}{2704156}\\ &= 0.0001065 \end{align*}$

ans: The probability that you win $500 is 0.0001065