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1986

JEE Main 1986 Mathematics question on Probability

A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II and III are pp, qq and 12\frac{1}{2}, respectively. If the probability that the student is successful, is 12\frac{1}{2}, then
  1. A.
    p=q=1p = q = 1
  2. B.
    p=q=12p = q = \frac{1}{2}
  3. C.
    p=1,q=0p = 1, q = 0
  4. D.
    p=1,q=12p = 1, q = \frac{1}{2}
Show answer and solution

Correct option: C

Solution

The event "successful" = (passes I and II) OR (passes I and III).

P(success) = P(I∩II) + P(I∩III) - P(I∩II∩III) = pq + p(1/2) - pq(1/2)

Setting this equal to 1/2:
p[q + 1/2 - q/2] = 1/2
p[q/2 + 1/2] = 1/2
p(q+1)/2 = 1/2
p(q+1) = 1

Since p and q are probabilities (0 ≤ p,q ≤ 1), the product p(q+1) = 1 requires p = 1 and q+1 = 1, giving q = 0. Options 1 and 4 don't satisfy this equation (e.g., p=1, q=1/2 gives 1·(3/2)=3/2 ≠ 1), and option 2 gives (1/2)(3/2)=3/4 ≠ 1. Only p=1, q=0 works.

[1986]

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