Question 1206321: 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(S2|A).
(Round your answer to four decimal places.)
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52754)   (Show Source):
You can put this solution on YOUR website! .
P(S2)*P(A|S2) P(S2)*P(A|S2)
P(S2|A) = ------------------- = -----------------------------------------------.
P(A) P(S1)*P(A|S1) + P(S2)*P(A|S2) + P(S3)*P(A|S3)
Substitute the numbers and get
P(S2|A) =  =  = 0.2778 (rounded). ANSWER
Solved.
Answer by math_tutor2020(3816)  (Show Source):
You can put this solution on YOUR website!
Answer: 0.2778 approximately
Work Shown
P(A) = P(A and S1)+P(A and S2)+P(A and S3) .... Law of Total Probability
P(A) = P(A given S1)*P(S1)+P(A given S2)*P(S2)+P(A given S3)*P(S3)
P(A) = 0.3*0.3 + 0.1*0.5 + 0.2*0.2
P(A) = 0.18
Event A has an 18% chance of occurring.
Then,
P(S2 given A) = P(S2 and A)/P(A)
P(S2 given A) = P(A given S2)*P(S2)/P(A) ..... Bayes Theorem
P(S2 given A) = 0.1*0.5/0.18
P(S2 given A) = 0.05/0.18
P(S2 given A) = 5/18
P(S2 given A) = 0.2778 approximately which is the answer.
The 7's go on forever, but of course you have to round to 4 decimal places.
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You might be thinking to yourself "what on earth just happened?"
It seems like those formulas come out of thin air.
Luckily there's a way to interpret things visually.
Get some graph paper.
Plot a square with one corner at (0,0) and the opposite corner at (10,10).
We have a 10 by 10 square.
See this question for a similar set up.
Draw vertical lines through x = 3 and x = 8
The horizontal span of 10 units is split into 3, 5, and 2 units when moving from left to right.
The three regions are labeled S1,S2,S3
Hopefully it doesn't take too much convincing that regions S1,S2,S3 take up 30%, 50%, and 20% of the original square.
P(S1) = 0.30 = 30%
P(S2) = 0.50 = 50%
P(S3) = 0.20 = 20%
For now focus entirely on the left-most section (S1).
This is the 3 by 10 rectangle.
Draw a horizontal line through y = 3.
Shade the lower part to represent P(A given S1) = 0.3
This indicates "if we are certain to be in region S1, then there's a 30% chance of being in the shaded blue region".
Note the height is 30% of the original. In other words, the blue area is 30% of the area of S1.
Now move to region S2.
For this region only, draw a horizontal line through y = 1 and shade the lower half. I'll do so in red.
If we are guaranteed that we landed in S2 somewhere, then there's a 10% chance of landing in the red area.
This represents P(A given S2) = 0.1
Now onto region S3.
Draw a horizontal line through y = 2 for this region only.
Shade the lower part. I'll do so in green.
The green area represents P(A given S3) = 0.2
How many shaded smaller pieces are contributed?
There's 3*3 = 9 blue pieces, 5*1 = 5 red pieces, and 2*2 = 4 green pieces.
That's a total of 9+5+4 = 18 shaded pieces overall.
Place them all in a bag (imagine they resemble poker chips of sorts). Shuffle the contents.
Blindly reach into the bag and select a piece. What are the chances of picking a red square?
Well there are 18 pieces total and 5 of which are red.
Therefore, 5/18 = 0.2778 is the final answer.
You have roughly a 27.78% chance of picking a red piece out of all of the shaded pieces total.
How are we certain there aren't any white squares in the bag?
It's all because of the fact we know 100% event A has occured.
Event A represents the total collection of all the shaded pieces.
Therefore, we are certain none of the pieces in the bag are white.
Put another way: let's say you threw a dart. You'll be blindfolded and a friend will mention "you landed in a blue region, a red region, or a green region. I won't tell you anything more". There are 18 such little squares you could have landed on. Of which 5 are from region S2.
Side note: Recall I mentioned that event A has an 18% chance of happening. This can be visually thought of having 18 shaded regions out of 100 total.
18/100 = 0.18 = 18%
If you have questions then please let me know.
More practice found at this page and this page
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