What is the probability that having 5 exams in 5 days a two

What is the probability that having 5 exams in 5 days

a) two or more are in the same day

b) have only one per day

c) all 5 the same day

Solution

the first exam can be on any one of the 5 days,the second exam can also be any on one of the 5 days,so the third exam and same for the 4th exam and the 5th exam.

so all possible cases=5*5*5*5*5

a) let A be the event that two or more are in the same day.

so P[A]=1-P[A^c]=1-P[all 5 are on different days]=1-P[the first exam can be on any one of the 5 days,the second exam can be on any one of the rest 4 days, the third exam can be on any one of the rest 3 days,so 2 days available for the 4th exam and 1 day available for the 5th exam]=1-[5*4*3*2*1/(5*5*5*5*5)]=0.9616 [answer]

b) let B be the event that only one exam per day.

so P[B]=P[the first exam can be on any one of the 5 days,the second exam can be on any one of the rest 4 days, the third exam can be on any one of the rest 3 days,so 2 days available for the 4th exam and 1 day available for the 5th exam]=[5*4*3*2*1/(5*5*5*5*5)]=0.0384 [answer]

c) let C be the event that all 5 exams are on the same day.

so P[C]=P[the first exam can be on any day of the 5 days,the rest 4 exams are on the same day which means that there are 5 choices for the first exam and only 1 choices for the rest 4 exams]

           =5*1*1*1*1/(5*5*5*5*5)=1/5⁴=0.0016 [answer]