Answer:
The probability for winning the grand prize is 0.000000104
Step-by-step explanation:
Probability is defined as the ratio of number of favorable outcomes to the number of possible outcomes.
Mathematically, we can write
[tex]\text{Probability}=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}[/tex]
It has been given that a player selects 7 numbers out of the first 80 positive integers. Here, the order doesn't matter so it is a case of combination.
Formula for combination is given by
[tex]^{n}C_r=\frac{n!}{r!(n-r)!}[/tex]
So, Number of possible outcomes is given by
[tex]^{80}C_7[/tex]
Using the formula for combination, we get
[tex]^{80}C_7=\frac{80!}{7!(80-7)!}\\\\=\frac{80!}{7!\cdot73!}[/tex]
And the total number of favorable outcomes is given by
[tex]^{11}C_7=\frac{11!}{7!(11-7)!}\\\\=\frac{11!}{7!\cdot4!}[/tex]
Therefore, probability for winning the grand prize is given by
[tex]\text{Probability}=\frac{\frac{80!}{7!\cdot73!}}{\frac{11!}{7!\cdot4!}}[/tex]
On simplifying, we get
[tex]\text{Probability}=\frac{330}{3.1767164\times10^9}\\\\\text{Probability}=0.000000104[/tex]
