Question 1172434: Use the contingency table to the right to determine the probability of events.
a. What is the probability of event A?
b. What is the probability of event A'?
c. What is the probability of event A and B?
d. What is the probability of event A or B?
Table_Data B B`
A 80 80
A' 50 70
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website! Part (a)
Given table
Compute the totals for each row and column. Then find the grand total to be 280, which is in the bottom right corner.
| B | B’ | Total | | A | 80 | 80 | 160 | | A’ | 50 | 70 | 120 | | Total | 130 | 150 | 280 |
If we divide everything by the grand total 280, and reduce fully, we get this table of probabilities
| B | B’ | Total | | A | 2/7 | 2/7 | 4/7 | | A’ | 5/28 | 1/4 | 3/7 | | Total | 13/28 | 15/28 | 1 |
From the first row, we see
P(A and B) = 2/7
P(A and B') = 2/7
Which means:
P(A) = P(A and B) + P(A and B')
P(A) = 2/7 + 2/7
P(A) = 4/7
This matches with the total shown in the table for row 1.
Answer: 4/7
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Part (b)
Apply the law of probability complements
P(A) + P(A') = 1
P(A') = 1 - P(A)
P(A') = 1 - (4/7)
P(A') = (7/7) - (4/7)
P(A') = (7-4)/7
P(A') = 3/7
This matches with the total for the A' row.
As an alternative approach:
P(A') = P(A' and B) + P(A' and B')
P(A') = 5/28 + 1/4
P(A') = 5/28 + 7/28
P(A') = 12/28
P(A') = 3/7
Answer: 3/7
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Part (c)
Refer to the probability table in part (a). Look in the upper left corner which corresponds to the A row and B column.
Answer: 2/7
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Part (d)
P(B) = P(B and A) + P(B and A')
P(B) = 2/7+5/28
P(B) = 8/28+5/28
P(B) = 13/28
This matches with the total shown at the bottom of column B in the probability table back in part (a).
P(A or B) = P(A) + P(B) - P(A and B)
P(A or B) = 4/7 + 13/28 - 2/7
P(A or B) = 16/28 + 13/28 - 8/28
P(A or B) = (16+13-8)/28
P(A or B) = 21/28
P(A or B) = 3/4
Answer: 3/4
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