For a certain probability experiment, the probability that event A wil

Question

For a certain probability experiment, the probability that event A will occur is \(\frac{1}{2}\) and the probability that event B will occur is \(\frac{1}{3}\). Which of the following values could be the probability that the event A ∪ B (that is, the event A or B, or both) will occur?

I. \(\frac{1}{3}\)

II. \(\frac{1}{2}\)

III. \(\frac{3}{4}\)

A. I only
B. II only
C. III only
D. II and III only
E. I, II and III

I.  \frac{1}{3}

II.  \frac{1}{2}

III.  \frac{3}{4}

A. I only
B. II only
C. III only
D. II and III only
E. I, II and III

CrackverbalGMAT · Accepted Answer

Sometimes the way the question stem is framed gives you a clue on how you need to proceed. The question says, 'which of the following COULD BE TRUE'. This means that we can have multiple possibilities, so instead of trying to get a definite answer, it would be easier to find the maximum and minimum possibilities and get a range.

This is a tricky question where you need to be very careful about making assumptions. Keep in mind that we have not been given any information on the relationship between the two events A and B. Since we have not been given the relationship between the events, the only thing that we can compute here are the maximum and the minimum possibilities.

Now while solving probability questions using the word 'OR', the best way to solve will be to represent the probabilities as a Venn Diagram.

There will be two scenarios here:

1. Mutually Exclusive Events : There will be no intersection between the two circles of the Venn. So the total probability will be the the addition of the two individual probabilities. This will give you the maximum probability that the two events will occur. So the max value here will be 1/3 + 1/2 = 5/6.

2. Mutually Inclusive Events : There will be an intersection between the two circles of the Venn. If the first circle represents the probability of A, the second represents the probability of B, then the Probability (A or B) = P(A) + P(B) - P(A and B). The minimum probability here will be when the B is a subset of A (since A is greater). So P(A ∪ B) is just the probability of A i.e. 1/2.

Since the min value is 1/2 and the max value is 5/6, any value in between the two (inclusive of the two) will be possible values. So II and III will work.

Answer : D

Aditya
CrackVerbal Quant Expert

nick1816 · Answer

The minimum value of A∪B is when B is subset of A. Hence minimum value of A ∪ B is 1/2.
The maximum value of A∪B is when A and B are two disjoint events, then A ∪ B is 1/2+1/3=5/6

Hence 0.5 ≤ A ∪ B ≤ 0.83
Only 1/2 and 3/4 lies in the interval. IMO D

Archit3110 · Answer

P A = 1/2 =0.5
P B = 1/3 ; 0.33
so P A or P B ; 1/2+1/3 ;5/6 ; .83
range is 0.5 to 0.83
so answer option 1/2 & 3/4 since it will cover all the aforementioned ranges 
IMO D

ScottTargetTestPrep · Answer

The smallest value of P(A U B) is ½ (occurring when B is a subset of A). The largest value of P(A U B) is ½ + ⅓ = ⅚ (occurring when A and B are disjoint). Therefore, ½ ≤ P(A U B) ≤ ⅚.

Answer: D

Kinshook · Answer

For a certain probability experiment, the probability that event A will occur is  \frac{1}{2}  and the probability that event B will occur is  \frac{1}{3} .

Which of the following values could be the probability that the event A ∪ B (that is, the event A or B, or both) will occur?

P(A) = 1/2
P(B) = 1/3

P(AUB) is equal to or more than 1/2 and is less than or equal to 1 - 1/2*1/3 = 5/6.

I.  \frac{1}{3}

II.  \frac{1}{2}

III.  \frac{3}{4}

IMO D

KarishmaB · Answer

P(A ∪ B) = P(A) + P(B) - P(Both)

P(A ∪ B) = \frac{1}{2} + \frac{1}{3} - P(Both)

P(A ∪ B) = \frac{5}{6} - P(Both)

Since probabilities are 1/2 and 1/3, the maximum value of P(Both) is 1/3. 
The minimum value of P(Both) = 0 since they could easily be mutually exclusive events (since 1/2 + 1/3 < 1).

P(A ∪ B) = \frac{5}{6} - \frac{1}{3}= \frac{1}{2}  -> Minimum Value
 P(A ∪ B) = \frac{5}{6} - 0= \frac{5}{6}  -> Maximum Value

1/3 does not lie in this range but the other two do.

Answer (D)

For a certain probability experiment, the probability that event A wil

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For a certain probability experiment, the probability that event A wil

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