Original practice · fully worked
Original variant: conditional alert attribution from a joint model
A monitoring system reports X critical faults, where X is 0, 1, or 2, and a rerouting flag Y, where Y is 0 or 1. Their joint pmf is p(x,y)=c(x+2y+1) on these six pairs. Given X+Y≥2, calculate P(Y=1).
- A
1/4 - B
1/3 - C
1/2 - D
2/3 - E
3/4
Answer and calculation
The conditioning event contains weights 3, 4, and 5 for pairs (2,0), (1,1), and (2,1). The two flagged pairs have total weight 9 of 12, so the conditional probability is 3/4 and choice E.
Setup
Setup
List the pairs satisfying the threshold and their unnormalized joint weights.
(2,0):3c,(1,1):4c,(2,1):5c Model
Model
Within the conditioning event, the flag is present for the last two pairs. The common normalizing constant cancels.
Pr(Y=1∣X+Y≥2)=3c+4c+5c4c+5c Compute
Compute
Sum the favorable and conditioning weights and reduce the ratio.
Pr(Y=1∣X+Y≥2)=129=43 Answer
Answer
The rerouting flag is present with conditional probability 3/4.
43(E) 