Sample 2020-21 Class 12 Maths Q15 Solution: Probability that a problem is solved when A and B try it independently

The question

The probabilities of A and B solving a problem independently are $\frac{1}{3}$ and $\frac{1}{4}$ respectively. If both of them try to solve the problem independently, what is the probability that the problem is solved?

Exam answer

Let $E$ and $F$ be the events that A and B solve the problem. Then $P(E) = \frac{1}{3}$ ⁠, $P(F) = \frac{1}{4}$ ⁠.

$P(\bar E) = 1 - \frac{1}{3} = \frac{2}{3}$ and $P(\bar F) = 1 - \frac{1}{4} = \frac{3}{4}$ ⁠.

Why?

$\bar E$ is the event that A does not solve the problem. An event and its complement have probabilities adding to 1.

\begin{aligned} 1 - \frac{1}{3} &= \frac{3}{3} - \frac{1}{3} = \frac{2}{3} \\ 1 - \frac{1}{4} &= \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \end{aligned}

E and F are independent, so $\bar E$ and $\bar F$ are independent.

\begin{aligned} P(\text{solved}) &= 1 - P(\bar E)P(\bar F) \\ &= 1 - \frac{2}{3} \times \frac{3}{4} \\ &= 1 - \frac{1}{2} \end{aligned}

Why?

The problem is solved when at least one of them solves it. The only other case is that neither solves it, so

P(\text{solved}) = 1 - P(\bar E \cap \bar F)

For independent events the probability of both is the product. Then $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$ ⁠.

\begin{aligned} \frac{2}{3} \times \frac{3}{4} &= \frac{2 \times 3}{3 \times 4} = \frac{6}{12} \\ &= \frac{6 \div 6}{12 \div 6} = \frac{1}{2} \\[4pt] 1 - \frac{1}{2} &= \frac{2}{2} - \frac{1}{2} = \frac{1}{2} \end{aligned}

Watch out: $\frac{1}{3} + \frac{1}{4} = \frac{7}{12}$ is wrong, because it counts the case "both solve it" twice.

$\therefore$ the probability that the problem is solved is $\dfrac{1}{2}$ ⁠.

Where this comes from

Chapter 13, Probability: Independent events: probability of at least one.

Question type: At least one, or exactly one

How to spot it: click to see

Type 6. At least one, or exactly one: think "not" first

Spot it: independent events, with "at least one", "the problem is solved", "neither" or "exactly one".

Here: A and B work independently, and "the problem is solved" means at least one of them solves it. So take 1 minus the chance both fail: $1 - \frac23\times\frac34 = \frac12$ ⁠.

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Q15: Probability that a problem is solved when A and B try it independently