SOLUTION: A sample is selected from one of two populations,
S1 and S2,
with
P(S1) = 0.7
and
P(S2) = 0.3.
The probabilities that an event A occurs, given that event
S1
or
S2
Algebra ->
Probability-and-statistics
-> SOLUTION: A sample is selected from one of two populations,
S1 and S2,
with
P(S1) = 0.7
and
P(S2) = 0.3.
The probabilities that an event A occurs, given that event
S1
or
S2
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Question 1206320: A sample is selected from one of two populations,
S1 and S2,
with
P(S1) = 0.7
and
P(S2) = 0.3.
The probabilities that an event A occurs, given that event
S1
or
S2
has occurred are
P(A|S1) = 0.4
and
P(A|S2) = 0.3
and the probability of event A is
P(A) = 0.37.
Use Bayes' Rule to find
P(S2|A).
(Round your answer to four decimal places.) Found 2 solutions by ikleyn, Edwin McCravy:Answer by ikleyn(52756) (Show Source):
We are given P(A|S2) = 0.3.
It means that P(A and S2) = 0.3*P(S2) = 0.3*0.3 = 0.09.
Next,
P(S2 and A)
P(S2|A) = ------------- by the definition of the conditional probability.
P(A)
In this formula, we just know the numerator P(S2 and A) = 0.09 (found above)
and the denominator P(A) = 0.37 (given).
Therefore, by substituting these values into the formula, we obtain
P(S2|A) = = 0.2432 (rounded).
ANSWER. P(S2|A) = 0.2432 (rounded).
While this can done using the conditional probability equation,
as Ikleyn has shown, we were asked to use Bayes' Rule, (which is
proved by using the conditional probability equation). Bayes' rule
is this proportion:
Substituting:
That rounds to 0.2432
Edwin
