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GO Classes DPP | GATE DA | Probability | Conditional Probability

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3 3 votes

In a certain city, the probability that a person will carry an umbrella on a given day is $0.5$. It is known that if the morning is cloudy, then the probability that the person will carry an umbrella is $0.7$. The morning is equally likely to be cloudy or not cloudy. What is the probability that the person will carry an umbrella on a day when the morning is not cloudy?

  1. $0.2$
     
  2. $0.3$
     
  3. $0.4$
     
  4. $0.5$

2 Answers

1 1 vote

Let $U$ denote the event that the person carries an umbrella, and let $C$ denote the event that the morning is cloudy.

Given:
$P(U) = 0.5$, $P(U \mid C) = 0.7$, and $P(C) = 0.5$.

Since the morning is equally likely to be cloudy or not cloudy,

$\therefore P(C^c) = 1 - 0.5 = 0.5$.

Let $P(U \mid C^c) = x$.

By the law of total probability,

$P(U) = P(U \mid C)P(C) + P(U \mid C^c)P(C^c)$.

Hence,
$0.5 = (0.7)(0.5) + x(0.5)$

$0.5 = 0.35 + 0.5x$

$0.15 = 0.5x$

$x = 0.3$

$\therefore P(U \mid C^c) = \boxed{0.3}$.

So the correct option is B.

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