A local game show has participants randomly select container

A local game show has participants randomly select containers, some of which have prizes in them. There are 30 containers total and only five prizes. Each prize obtained raises the overall level of cash the contestant wins, with the grand prize given only to a contestant that finds all five prizes. If a contestant is allowed to make eight selections, what is the probability that he or she will win the grand prize?

Solution

there are 30 containers and only 5 prizes.

grand prize is given only to a contestant that finds all five prizes.

so out of 30 containers, 5 containers contain prizes and 25 containers do not contain prizes.

a contestant is allowed to make 8 selections.

let A be the event that he or she will win the grand prize

so P[A]=probability that he/she chooses 5 containers from the 5 containers that contain prizes and 3 containers from the remaining 25 containers that do not contain prizes.

now he/she can choose 5 containers from the 5 containers that contain prizes in ⁵C₅ ways.

and 3 containers from the remaining 25 containers that do not contain prizes in ²⁵C₃ ways.

and he/she can choose overall 8 containers from 30 containers in ³⁰C₈ ways.

so P[A]=(cases favourable to A)/(total number of cases)=⁵C₅*²⁵C₃/³⁰C₈=0.000393 [answer]