Question 1206319: 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) = 0.2
and
P(A|S2) = 0.1.
Use the Law of Total Probability to find P(A).
Found 4 solutions by Theo, ikleyn, Edwin McCravy, math_tutor2020:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i think this works like this.
law of total probability states that p(A) = p(A intersect B) plus p(A intersect complement of B.
you are given:
p(S1) = .7
p(S2) = .3
you are given that p(A given S1) = .2.
you are given that p(A given S2) = .1
p(A given S1) is equal to p(A intersect S1) / p(S1).
this becomes .2 = p(A intersect S1) / .7
solve for p(A intersect S1) to get:
p(A intersect S1) = .7 * .2 = .14
p(A given S2) is equal to p(A intersect S1) / p(S2).
this becomes .1 = p(A intersect S1) / .3
solve for p(A intersect S2) to get:
p(A intersect S2) = .1 * .3 = .03
p(A) is equal to p(A intersect S1) + p(A intersect S2) = .14 + .03 = .17
that should be your answer.
here's a reference.
https://www.statisticshowto.com/total-probability-rule/
i drew a tree diagram (not very pretty) that shows the same result.
see below.
in the diagram, that's p(S1 and S2) and p(A and not A).
the top of the tree is marked as 1, meaning it's 100% probability on top of the tree.
that encompasses all that's beneath it.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
As with other today's posts (and, generally, with many other posts at this forum),
the problem formulation is INCORRECT.
It is incorrect, because it is incomplete.
To be complete, it must say in the condition, that populations S1 and S2 are disjoint (= mutually exclusive).
The fact that P(S1) + P(S2) = 0.7 + 0.3 = 1 does not imply automatically that S1 and S2 are disjoint,
since they still may have intersection (for example, S2 can be a part of S1).
///////////////////
Edwin, I do not read in the head of the instructor.
I even do not try to do it . . .
I read what I see in the post.
Answer by Edwin McCravy(20055)  (Show Source):
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Answer: 0.17
Work Shown
P(A) = P(A and S1) + P(A and S2)
P(A) = P(A given S1)*P(S1) + P(A given S2)*P(S2)
P(A) = 0.2*0.7 + 0.1*0.3
P(A) = 0.14 + 0.03
P(A) = 0.17
Event A has a 17% chance of happening.
In reference to tutor Theo's drawing, the left-most branch represents the P(A given S1)*P(S1) portion.
P(S1) is up top and P(A given S1) is the node underneath to the left.
The other branch that ends with "A" follows the pathway from S2 to A.
That would be the P(A given S2)*P(S2) portion.
Here's another diagram we could make.
Draw a 10 by 10 square. Split it into two pieces labeled S1 and S2
S1 takes up 70% of the area
S2 takes up 30% of the area
P(S1) = 0.7
P(S2) = 0.3
Similar diagrams can be found here and here
Then we shade in the blue and red areas shown below.
The blue area is 20% of the area of S1 (i.e. the blue height is 20% of the height of the original square).
If we are certain to land in S1, then there is a 20% chance of landing in the blue area.
That leads to P(A given S1) = 0.20 = 20%
The red area is 10% of the area of S2
P(A given S2) = 0.10 = 10%
Either shaded region represents event A.
The blue region is 7 units across and 2 units tall.
There are 7*2 = 14 blue squares.
There are also 3*1 = 3 red squares
In all there are 14+3 = 17 squares shaded either blue or red.
This is out of 10*10 = 100 squares overall.
17/100 = 0.17 is the answer we arrived at earlier.
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