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ACT Exam  >  ACT Questions  >  Suppose P(A) = 0.4, P(B) = P and P(A∪ B) ... Start Learning for Free
Suppose P(A) = 0.4, P(B) = P and P(A ∪ B) = 0.7. If A and B are independent events, then the value of P is:
  • a)
    0.5
  • b)
    0.3
  • c)
    0.55
  • d)
    0.6
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Suppose P(A) = 0.4, P(B) = P and P(A∪ B) = 0.7. If A and B are ind...
Concept:
  • For two events A and B, we have: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
  • If A and B are independent events, then P(A ∩ B) = P(A) × P(B).
Calculation:
Using the concept above, because A and B are independent events, we can write:
P(A ∪ B) = P(A) + P(B) - P(A) × P(B)
⇒ 0.7 = 0.4 + P - 0.4 × P
⇒ 0.6P =0.3
⇒ P = 0.5.
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Most Upvoted Answer
Suppose P(A) = 0.4, P(B) = P and P(A∪ B) = 0.7. If A and B are ind...
Concept:
  • For two events A and B, we have: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
  • If A and B are independent events, then P(A ∩ B) = P(A) × P(B).
Calculation:
Using the concept above, because A and B are independent events, we can write:
P(A ∪ B) = P(A) + P(B) - P(A) × P(B)
⇒ 0.7 = 0.4 + P - 0.4 × P
⇒ 0.6P =0.3
⇒ P = 0.5.
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Community Answer
Suppose P(A) = 0.4, P(B) = P and P(A∪ B) = 0.7. If A and B are ind...
Understanding the Problem
We are given the probabilities of two events A and B, along with the probability of their union. The values provided are:
- P(A) = 0.4
- P(B) = P (unknown)
- P(A ∪ B) = 0.7
Additionally, events A and B are independent.
Formula for Union of Events
For any two events A and B, the probability of their union is given by:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Since A and B are independent events, we can express P(A ∩ B) as:
P(A ∩ B) = P(A) * P(B)
Substituting Known Values
Substituting the known values into the union formula:
0.7 = P(A) + P(B) - P(A) * P(B)
Now, substituting P(A):
0.7 = 0.4 + P - (0.4 * P)
Simplifying the Equation
Rearranging gives:
0.7 = 0.4 + P - 0.4P
This simplifies to:
0.7 = 0.4 + P(1 - 0.4)
0.7 = 0.4 + 0.6P
Now, isolate P:
0.6P = 0.7 - 0.4
0.6P = 0.3
P = 0.3 / 0.6
P = 0.5
Conclusion
Thus, the value of P is 0.5, which matches the answer option 'A'.
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Adapted from"Why a Classic is a Classic" inLiterary Taste: How to Form Itby Arnold Bennet(1909)The large majority of our fellow-citizens care as much about literature as they care about airplanes or the policies of the Legislature. They do not ignore it; they are not quite indifferent to it. But their interest in it is faint and perfunctory; or, if their interest happens to be violent, it is spasmodic. Ask the two hundred thousand persons whose enthusiasm made the vogue of a popular novel ten years ago what they think of that novel now, and you will gather that they have utterly forgotten it, and that they would no more dream of reading it again than of reading Bishop StubbssSelect Charters. Probably if they did read it again they would not enjoy it—not because the said novel is worse now than it was ten years ago; not because their taste has improved—but because they have not had sufficient practice to be able to rely on their taste as a means of permanent pleasure. They simply dont know from one day to the next what will please them.In the face of this one may ask: Why does the great and universal fame of classical authors continue? The answer is that the fame of classical authors is entirely independent of the majority. Do you suppose that if the fame of Shakespeare depended on the man in the street it would survive a fortnight? The fame of classical authors is originally made, and it is maintained, by a passionate few. Even when a first-class author has enjoyed immense success during his lifetime, the majority have never appreciated him as sincerely as they have appreciated second-rate men. He has always been reinforced by the ardor of the passionate few. And in the case of an author who has emerged into glory after his death the happy sequel has been due solely to the obstinate perseverance of the few. They could not leave him alone; they would not. They kept on savoring him, and talking about him, and buying him, and they generally behaved with such eager zeal, and they were so authoritative and sure of themselves, that at last the majority grew accustomed to the sound of his name and placidly agreed to the proposition that he was a genius; the majority really did not care very much either way.And it is by the passionate few that the renown of genius is kept alive from one generation to another. These few are always at work. They are always rediscovering genius. Their curiosity and enthusiasm are exhaustless, so that there is little chance of genius being ignored. And, moreover, they are always working either for or against the verdicts of the majority. The majority can make a reputation, but it is too careless to maintain it. If, by accident, the passionate few agree with the majority in a particular instance, they will frequently remind the majority that such and such a reputation has been made, and the majority will idly concur: "Ah, yes. By the way, we must not forget that such and such a reputation exists." Without that persistent memory-jogging the reputation would quickly fall into the oblivion which is death. The passionate few only have their way by reason of the fact that they are genuinely interested in literature, that literature matters to them. They conquer by their obstinacy alone, by their eternal repetition of the same statements. Do you suppose they could prove to the man in the street that Shakespeare was a great artist? The said man would not even understand the terms they employed. But when he is told ten thousand times, and generation after generation, that Shakespeare was a great artist, the said man believes--not by reason, but by faith. And he too repeats that Shakespeare was a great artist, and he buys the complete works of Shakespeare and puts them on his shelves, and he goes to see the marvelous stage-effects which accompanyKing LearorHamlet, and comes back religiously convinced that Shakespeare was a great artist. All because the passionate few could not keep their admiration of Shakespeare to themselves. This is not cynicism; but truth. And it is important that those who wish to form their literary taste should grasp it.Q. Why does the author believe the majority can be convinced of Shakespeare’s genius?

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Suppose P(A) = 0.4, P(B) = P and P(A∪ B) = 0.7. If A and B are independent events, then the value of P is:a)0.5b)0.3c)0.55d)0.6Correct answer is option 'A'. Can you explain this answer?
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