Explanation

Since you know that the probability of event A is \(\frac{1}{2}\)and the probability of event B is \(\frac{1}{3}\) but you are not given any information about the relationship between events A and B, you can compute only the minimum possible value and the maximum possible value of the probability of the event A ∪ B. The probability of A ∪ B is least if B is a subset of A; in that case, the probability of A ∪ B is just the probability of A, or \(\frac{1}{2}\) .

The probability of A ∪ B is greatest if A and B do not intersect at all; in that case, the probability of A ∪ B is the sum of the probabilities of A and B, or \(\frac{1}{2} + \frac{1}{3} = \frac{5}{6}\) With no further information about A and B, the probability that A or B, or both, will occur could be any number from \(\frac{1}{2}\) to \(\frac{5}{6}\).
Of the answer choices given, only \(\frac{1}{2}\) and \(\frac{3}{4}\) are in this interval.

The correct answer consists of Choices B and C.
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This is the correct expression

P(A ∪ B) = P(A) + P(B) - P(B)P( A | B )

P(A) is probability of A happening
P(B) is probability of B happening
P( B | A ) can be said to Probability of B given that A has occured. Now look at the figure below. The area P( A | B ) is minimum when P(A) and P(B) do not overlap at all.

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Intutive explanation of the formula:

P(A ∪ B) = P(A) + P(B) - P(B)P( A | B )

If you look at the figure to find area of P(A ∪ B) we add the area of P(A) and P(B) but if P(A) and P(B) overlap, we would add the overlapping area twice! So we sunbtract P(B)P( A | B ).

Example:

An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn without replacement from the urn. What is the probability that both of the marbles are black?


Let A = the event that the first marble is black; and let B = the event that the second marble is black.

We know the following:

In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.
After the first selection, there are 9 marbles in the urn, 3 of which are black. Therefore, P(B|A) = 3/9.

P(A) P(B|A) = (4/10) * (3/9) = 12/90 = 2/15 = 0.133


P(B)P( A | B ) is not P(B)P(A). The amount of overlapping area cannot be found out unless specified

Please take care of this as this is the very fundamental concept of probability. Hope this helps and feel free to ask more questions.
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Sandy
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The answers are B and C.

It is an official question.

Regards
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Probability of event \(B\) is \(0.5\)
Probability of event \(A\) is \(0.3333\)

Given that probability of event \(B\) is greater than \(A\)
one A U B scenario could involve event \(A\) being a subset of \(B\) hence minimum possible union will be \(0.5\)
However if the two probabilities are mutually exclusive \(A + B\) would be \(0.8333\) which would be the maximum probability
as such the range of probability for A U B is \(0.5\) to \(0.8333\) inclusive
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This is my response to the question and may be incorrect. Feel free to rectify any mistakes