CBSE 2025, Set 3

Question

In a city, a survey was conducted among residents about their preferred mode of commuting. It was found that $50\%$ people preferred using public transport, $35\%$ preferred using a bicycle and $20\%$ use both public transport and a bicycle. If a person is selected at random, find the probability that :
(i) The person uses only public transport.
(ii) The person uses a bicycle, given that they also use the public transport.
(iii) The person uses neither public transport nor a bicycle.

Edvaya · Accepted Answer

The correct answer is Subjective. Let $A$ be the event that a person uses public transport.
 Let $B$ be the event that a person uses a bicycle.
 Given the probabilities:
 $P(A) = 50\% = 0.50$
 $P(B) = 35\% = 0.35$
 $P(A \cap B) = 20\% = 0.20$

**(i) The person uses only public transport:**
 This means they use public transport but NOT a bicycle. This is represented by $P(A \cap B')$.
 $$P(	ext{only } A) = P(A) - P(A \cap B)$$
 $$P(	ext{only } A) = 0.50 - 0.20 = 0.30$$
 Therefore, the probability is $0.30$ (or $30\%$).

**(ii) The person uses a bicycle, given that they use public transport:**
 This requires conditional probability $P(B|A)$.
 $$P(B|A) = \frac{P(A \cap B)}{P(A)}$$
 $$P(B|A) = \frac{0.20}{0.50} = \frac{2}{5} = 0.40$$
 Therefore, the probability is $0.40$ (or $40\%$).

**(iii) The person uses neither public transport nor a bicycle:**
 First, we find the probability that a person uses *at least one* of the modes, which is $P(A \cup B)$.
 Using the addition theorem of probability:
 $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
 $$P(A \cup B) = 0.50 + 0.35 - 0.20 = 0.65$$

The probability of using neither is the complement of using at least one:
 $$P(	ext{neither}) = 1 - P(A \cup B)$$
 $$P(	ext{neither}) = 1 - 0.65 = 0.35$$
 Therefore, the probability is $0.35$ (or $35\%$).

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