Question

The probability that a new car will get an award for its design is $0.25$, the probability that it will get an award for efficient use of fuel is $0.35$ and the probability that it will get both the awards is $0.15$. Find the probability that:
(i) It will get at least one of the two awards.
(ii) It will get only one of the awards.

Answer 1

Let P ($A$) be the probability that a new car will get an award for its design
P ($A$) = $0.25$
Let P ($B$) be the probability that it will get an award for efficient use of fuel
P ($B$) = $0.35$
Let P $(A\cap B)$ be the probability that it will get both the awards
P $(A\cap B)$ = $0.15$
(i) it will get at least one of the two awards
P $(A\cup B)$
= P ($A$) + P ($B$) - P $(A\cap B)$
= $0.25+0.35-0.15$
= $0.45$
(ii) it will get only one of the awards
= P ($A^{\prime }\cap B$) + P ($A\cap B^{\prime }$)
= P($A$) - P ($A\cap B$) + P(B) - ($A\cap B$)
= $(0.25-0.15)+(0.35-0.15)$
= $0.10+0.20$
= $0.30$