Question

Let A and B be two independent events such that the odds in favour of A and B are \(1:1 \) and \(3:2\) respectively .Then the probability that only one of the two occurs is 

• A. \(0.6\)
• B. \(0.7\)
• C. \(0.8\)
• D. \(0.5\)
• E. \(0.4\)

Answer 1

The Correct Option is D

Given that:

Fir the given two events A and B

Odds in favor of A = 1:1.

Odds in favor of B = 3:2. 

  Therefore, probability of A  P(A) =\(\dfrac{1}{(1 + 1)} = \dfrac{1}{2}\)         and         probability of B  P(B) = \(\dfrac{3}{(3 + 2)} = \dfrac{3}{5} \).

As per the question we need to find the probability that only one of the two events occurs. This means either event A occurs (but not B), or event B occurs (but not A).

Probability of only A occurring = P(A) × (1 - P(B)) =\( (\dfrac{1}{2}) × (1 - \dfrac{3}{5})\)

 \(= (\dfrac{1}{2}) × (\dfrac{2}{5})\)

\( = \dfrac{1}{5}\)

Probability of only B occurring = P(B) × (1 - P(A)) = \((\dfrac{3}{5}) × (1 - \dfrac{1}{2}) \)

\(= (\dfrac{3}{5}) × (\dfrac{1}{2}) \)

\(= \dfrac{3}{10}\)

Now, to find the total probability that only one of the two events occurs, we can write,

Total probability = Probability of only A  + Probability of only B 

\( = \dfrac{1}{5} + \dfrac{3}{10}\)

\(= \dfrac{1}{2}\)

So, the probability that only one of the two events A or B occurs is  \(\dfrac{1}{2}\).

Hence the correct answer is \(0.5\)  (_Ans.)