Question

When an experiment is conducted, one and only one of three mutually exclusive events S1, S2, and S3 can occur, with

P(S1) = 0.3,

P(S2) = 0.5,
and
P(S3) = 0.2.
The probabilities that a fourth event A occurs, given that event S1, S2, or S3 has occurred, are
P(A|S1) = 0.3 P(A|S2) = 0.1 P(A|S3) = 0.2.
If event A is observed, find
P(S3|A).
(Round your answer to four decimal places.)
P(S3|A) =

Answer 1

`P(S_3|A) = (2)/(9)`

Step-by-step explanation:

Given

`P(S1) = 0.3`
`P(S_2) = 0.5`
`P(S_3) = 0.2`

`P(A|S_1) = 0.3`
`P(A|S_2) = 0.1`
`P(A|S_3) = 0.2`

Required

Determine
`P(S_3|A)`

Using Bayes' theorem:

`P(S_3|A) = (P(S_3) * P(A|S_3))/(P(S_1) * P(A|S_1) + P(S_2) * P(A|S_2) + P(S_3) * P(A|S_3))`

So, we have:

`P(S_3|A) = (0.2 * 0.2)/(0.3 * 0.3+ 0.5* 0.1 + 0.2 * 0.2)`

`P(S_3|A) = (0.04)/(0.18)`

Multiply by 100/100

`P(S_3|A) = (4)/(18)`

`P(S_3|A) = (2)/(9)`