Question

A student appears for test I, II, III. The student is successful if he passes either in test I and II or test I and III. The probabilities of the student passing in test I, II, and III are p, q and $\dfrac{1}{2}$ respectively. If the probability that the student is successful is $\dfrac{1}{2}$then,
$
  A.p = q = 1 \\
  B.p = q = \dfrac{1}{2} \\
  C.p = 1,q = 0 \\
  D.p = 1,q = \dfrac{1}{2} \\
 $

Answer 1

Hint- If A and B are two independent events for a random experiment, then the probability of simultaneous occurrence of two independent events will be equal to the product of their probabilities. Hence, $P(A \cap B) = P(A).P(B)$
Use the Addition theorem: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$


Complete step by step solution:
The student keeps attention here the word either is used and also ‘OR’ for union and ‘AND’ for the intersection.
Let us consider that A, B and C are events of passing in test I, test II and test III, respectively.
According to the question, $P\left[ {\left( {A \cap B} \right) \cup \left( {A \cap C} \right)} \right] = \dfrac{1}{2}$
Using addition theorem,
$
  P(A \cup B) = P(A) + P(B) - P(A \cap B) \\
  P(A \cap B) + P\left( {A \cap C} \right) - P((A \cap B) \cap (A \cap C)) = \dfrac{1}{2} \\
  P(A \cap B) + P\left( {A \cap C} \right) - P(A \cap B \cap C) = \dfrac{1}{2}{\text{ - - - - (i) }}\left[ {P(A \cap A) = A} \right] \\
 $

Here, A, B and C are independent events so,

$P(A \cap B) = P(A).P(B)$ And, $P(A \cap C) = P(A).P(C)$ And, $P(A \cap B \cap C) = P(A).P(B).P(C)$

So, from equation (i), we get:
$
  P(A).P(B) + P(A).P(C) - P(A).P(B).P(C) = \dfrac{1}{2} \\
  pq + p(\dfrac{1}{2}) - pq(\dfrac{1}{2}) = \dfrac{1}{2} \\
  2pq + p - pq = 1 \\
  pq + p = 1 \\
  p(1 + q) = 1 \\
  p = 1;1 + q = 1 \\
  p = 1;q = 0 \\
 $

Option (C) is correct.

Additional Information: Probability means possibility; it is a branch of mathematics that deals with the occurrence of the event. It is used to predict how likely events are to happen. To find the probability of a single event to occur, first, we should know the total number of possible outcomes.
Probability can range in from 0 to 1, where 0 means the event to be an impossible one, and 1 indicates a certain event. The random experiment is an experiment where we know the set of all possible outcomes but find it impossible to predict one at any particular execution. If the probability of occurrence of an event A is not affected by the occurrence of another event B, then A and B are said to be independent events.

Note: In this question, the student should use the addition formula carefully. The main concept is to use a union for ‘OR’ and intersection for ‘AND’. In this question, there is the use of independent events concept.