Question 1172449: Use the contingency table below to find the following probabilities.
a. A|B
b. A|B'
c. A'|B'
d. Are events A and B independent?
Table_Data,B,B`
A,30,40
A',40,50
Answer by math_tutor2020(3817)   (Show Source):
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Original Table
Compute the subtotals and the grand total
| B | B' | Total | | A | 30 | 40 | 70 | | A' | 40 | 50 | 90 | | Total | 70 | 90 | 160 |
Divide each item by the grand total 160, and fully reduce, to compute the probabilities
| B | B' | Total | | A | 3/16 | 1/4 | 7/16 | | A' | 1/4 | 5/16 | 9/16 | | Total | 7/16 | 9/16 | 1 |
We'll use this probability table to answer parts (a) through (d)
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Part (a)
P(A and B) = 3/16 .... upper left corner of table
P(B) = 7/16 .... bottom of column 1
P(A | B) = Probability of A, given B
P(A | B) = P(A and B)/P(B)
P(A | B) = (3/16)/(7/16)
P(A | B) = (3/16)*(16/7)
P(A | B) = 3/7
Answer: 3/7
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Part (b)
P(A and B') = 1/4
P(B') = 9/16
P(A given B') = P(A and B')/P(B')
P(A given B') = (1/4) divide by (9/16)
P(A given B') = (1/4)*(16/9)
P(A given B') = (1*16)/(4*9)
P(A given B') = 16/36
P(A given B') = (4*4)/(4*9)
P(A given B') = 4/9
Answer: 4/9
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Part (c)
P(A' and B') = 5/16
P(B') = 9/16
P(A' given B') = P(A' and B')/P(B')
P(A' given B') = (5/16) divide by (9/16)
P(A' given B') = (5/16)*(16/9)
P(A' given B') = 5/9
Answer: 5/9
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Part (d)
Events A and B would be independent if and only if the following two items are true
P(A given B) = P(A)
P(B given A) = P(B)
Independent events are not linked together. If A and B are independent, then one event occurring does not change the probability of the other.
From part (a), we found P(A given B) = 3/7, but this is not the same value as P(A) = 7/16, which is what the table shows. This concludes that A and B are not independent.
As an alternative, we could also use the equation
P(A and B) = P(A)*P(B)
to find that
P(A and B) = P(A)*P(B)
3/16 = (7/16)*(7/16)
3/16 = 49/256
which is a false equation, so events A and B are not independent.
Answer: A and B are not independent. They are dependent.
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