Question 1191943: Use the following probabilities to answer the question. Don't round; enter your answers as ratios and let WAMAP do the calculations for you. It may be helpful to make a contingency table.
P(A)= 0.31, P(B)=0.76 and P( A and B)=0.18
P(B | A)=
p( Not B | A)=
P( Not B | Not A)=
Answer by math_tutor2020(3817)   (Show Source):
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Given info:
P(A) = 0.31
P(B) = 0.76
P(A and B) = 0.18
Based on the given info, we can say
P(not A) = 1 - P(A) = 1 - 0.31 = 0.69
P(not B) = 1 - P(B) = 1 - 0.76 = 0.24
In short,
P(not A) = 0.69
P(not B) = 0.24
This is what the contingency table starts off looking like
| A | not A | Total | | B | P(A and B) | P(not A and B) | P(B) | | not B | P(A and not B) | P(not A and not B) | P(not B) | | Total | P(A) | P(not A) | 1 |
Fill in the given values, and the values found so far
| A | not A | Total | | B | 0.18 | P(not A and B) | 0.76 | | not B | P(A and not B) | P(not A and not B) | 0.24 | | Total | 0.31 | 0.69 | 1 |
In the "A" column, notice how we're missing one value in the middle.
Apply the law of total probability
P(A and B) + P(A and not B) = P(A)
0.18 + P(A and not B) = 0.31
P(A and not B) = 0.31 - 0.18
P(A and not B) = 0.13
Then along the first row, we can say a similar argument.
P(A and B) + P(not A and B) = P(B)
0.18 + P(not A and B) = 0.76
P(not A and B) = 0.76 - 0.18
P(not A and B) = 0.58
Lastly, we can look at row 2 or column 2 to find that final missing probability.
P(A and not B) + P(not A and not B) = P(not B)
0.13 + P(not A and not B) = 0.24
P(not A and not B) = 0.24 - 0.13
P(not A and not B) = 0.11
Or,
P(not A and B) + P(not A and not B) = P(not A)
0.58 + P(not A and not B) = 0.69
P(not A and not B) = 0.69 - 0.58
P(not A and not B) = 0.11
To summarize so far, we calculated:
P(A and not B) = 0.13
P(not A and B) = 0.58
P(not A and not B) = 0.11
This is what the completed contigency table looks like
| A | not A | Total | | B | 0.18 | 0.58 | 0.76 | | not B | 0.13 | 0.11 | 0.24 | | Total | 0.31 | 0.69 | 1 |
Optionally if you wanted, you could convert those decimal values to their fractional forms.
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Now to calculate the conditional probabilities
P(B | A) = P(B and A)/P(A)
P(B | A) = P(A and B)/P(A)
P(B | A) = 0.18/0.31
P(B | A) = 18/31
P(not B | A) = P(not B and A)/P(A)
P(not B | A) = P(A and not B)/P(A)
P(not B | A) = 0.13/0.31
P(not B | A) = 13/31
Or another route you could take is:
P(not B | A) = 1 - P(B | A)
P(not B | A) = 1 - 18/31
P(not B | A) = 31/31 - 18/31
P(not B | A) = (31-18)/31
P(not B | A) = 13/31
This alternative route works because P(B | A) and P(not B|A) are complementary events in the same way that P(B) and P(not B) are complementary.
P(not B | not A) = P(not B and not A)/P(not A)
P(not B | not A) = P(not A and not B)/P(not A)
P(not B | not A) = 0.11/0.69
P(not B | not A) = 11/69
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Answers:
P(B | A) = 18/31
P(not B | A) = 13/31
P(not B | not A) = 11/69
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