SOLUTION: (1 point) Suppose that P(A)=0.4, P(C∣A)=0.0055, and P(C′∣A′)=0.0105. Find P(A∣C)

Question 1192485: (1 point) Suppose that P(A)=0.4, P(C∣A)=0.0055, and P(C′∣A′)=0.0105. Find P(A∣C)
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

P(A) = 0.4
P(A') = 1 - P(A) = 1 - 0.4 = 0.6

P(C | A) = 0.0055
P(C' | A) = 1 - P(C | A) = 1 - 0.0055 = 0.9945

P(C' | A') = 0.0105
P(C | A') = 1-P(C' | A') = 1-0.0105 = 0.9895

To summarize so far, we have:
P(A) = 0.4
P(A') = 0.6
P(C|A) = 0.0055
P(C|A') = 0.9895

Now use the Law of Total Probability to be able to say the following
P(C) = P(C and A) + P(C and A')
P(C) = P(C | A)*P(A) + P(C | A')*P(A')
P(C) = 0.0055*0.4 + 0.9895*0.6
P(C) = 0.5959

Next, apply Bayes Theorem
P(A | C) = P(C | A)*P(A)/P(C)
P(A | C) = 0.0055*0.4/0.5959
P(A | C) = 0.0036918946132
The result is approximate.
Round that value however you need to.

Here's one useful calculator to check your work.
https://www.gigacalculator.com/calculators/bayes-theorem-calculator.php
Type in these values
P(A) = 0.4
P(C|A) = 0.0055
P(C|A') = 0.9895
Make sure to click the radio button labeled "Proportions e.g. 0.05" before hitting "calculate".

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