A language class has only 3 students A,B and C and they independently attend the class. The probabilities of the attendance of A, B and C on a given day are 1/2, 1/3 and 3/4 respectively. Find the probability that the total number of attendance in 2 consecutive days is exactly 3.
Answers
Given : A language class has only 3 students . he probabilities of the attendance of A, B and C on a given day are 1/2, 1/3 and 3/4 respectively.
To find : probability that the total number of attendance in 2 consecutive days is exactly 3.
Solution:
total number of attendance in 2 consecutive days is exactly 3.
This can happen as below
1st day - 0 2nd day 3
1st day 1 2nd day 2
1st Day 2 2nd Day 1
1st day 3 2nd day 0
1st day - 0 2nd day 3
= (1 - 1/2)(1 - 1/3)(1 - 3/4) (1/2)(1/3)(3/4)
= (1/2)(2/3)(1/4) (1/8)
= 1/96
1st day 3 2nd day 0 will be same = 1/96
1st day 1 2nd day 2
1st day 1 = (1/2)(1 - 1/3)(1 - 3/4) + (1/3)(1 - 1/2)(1 - 3/4) + (3/4)( 1 - 1/2)(1 - 1/3)
= (1/2)(2/3)(1/4) + (1/3)(1/2)(1/4) + (3/4)(1/2)(2/3)
=1/12 + 1/24 + 1/4
= 9/24
= 3/8
2nd day 2 (1/2)(1/3)(1 - 3/4) + (1/2)(1 - 1/3)(3/4) + (1 - 1/2)(1/3)(3/4)
= (1/2)(1/3)(1/4) + (1/2)(2/3)(3/4) + (1/2)(1/3)(3/4)
= 1/24 + 1/4 + 1/8
= 10/24
= 5/12
1st day 1 2nd day 2 = (3/8)(5/12) = 15/96
1st Day 2 2nd Day 1 will be same = 15/96
probability that the total number of attendance in 2 consecutive days is exactly 3. =
2 ( 1/96 + 15/96)
= 2 ( 16/96)
= 32/96
= 1/3
1/3 is the probability that total number of attendance in 2 consecutive days is exactly 3.
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