Can someone please help me with this question (a and b)? I need detailed explana
ID: 3070578 • Letter: C
Question
Can someone please help me with this question (a and b)? I need detailed explanation! Thank you so much!!!
A, B and C live in an apartment together. A gets up in the morning before B and C is up last. First to the kitchen, A encounters a sink full of dirty dishes and washes them with probability PA 1/2. Similarly, after A has left, on encountering dirty dishes. B washes them with probability PB = 1 /3. Likewise, C washes them with probability pc-1/4. (a) What is the probability that the dishes get done? b) If the dishes get done, what is the probability that C washed them?Explanation / Answer
Let A, B, C be the respective events of washing the dirty dishes and A' , B', C' be the respective events of not washing the dirty dishes . Then A, B, C are independent event
A', B', C' are independent events.
Now, P(A) = 1/2 , P(B) = 1/3 and P(C)=1/4
P(A')= 1- 1/2 = 1/2, P(B')=1 - 1/3 = 2/3, P(C')= 1 - 1/4 = 3/4.
P( dishes not get done) = P[(not A) and (not B) and (not C)]
= P(A'B'C')
= P(A')P(B')P(C') [ A, B, C are Independent]
= 1/2×2/3×3/4
= 1/4
Hence, P(the problem will be solved) = 1 - P(none solves the problem)
= 1141-14= 3/4
If the dishes get done, then the probability that C washed them
= P[(not A) and (not B) and (C)]
= P(A'B'C)
= P(A')P(B')P(C) [ A, B, C are Independent]
= 1/2×2/3×1/4
= 1/12