Q.Probability of solving specific problem independently by A and B are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that

Concept: Event Independence — For independent events, $P(A \cap B) = P(A) \cdot P(B)$.

Let $A$ = event that A solves the problem, $B$ = event that B solves the problem.

Given $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$, and $A, B$ are independent.

1. Probability the problem is solved (at least one solves): $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ Since independent, $P(A \cap B) = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6}$ So $P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{3+2-1}{6} = \frac{4}{6} = \frac{2}{3}$
2. Probability exactly one solves: $P(\text{exactly one}) = P(A \cap B^c) + P(A^c \cap B)$ $= P(A) \cdot P(B^c) + P(A^c) \cdot P(B)$ …

Let’s start with the core concept. When events are independent, the chance that both happen is simply the product of their individual probabilities. Here, A and B each try to solve the problem without influencing each other — that’s what “independently” means. So we can multiply their success and failure probabilities freely.

The problem gives:

• $P(A \text{ solves}) = \frac{1}{2}$
• $P(B \text{ solves}) = \frac{1}{3}$

From these, we immediately get the failure probabilities:

• $P(A \text{ fails}) = 1 - \frac{1}{2} = \frac{1}{2}$
• $P(B \text{ fails}) = 1 - \frac{1}{3} = \frac{2}{3}$

Now let’s tackle each part.

1. Probability that the problem is solved The problem is solved if at least one of them solves it. The easiest way is to use the complement: the only way the problem remains unsolved is if both fail. Since A and B work independently:

Therefore:

2. Probability that exactly one of them solves the problem …

Method: "Problem solved" and "exactly one" for independent attempts

When two people (or components) act independently with individual success probabilities, questions about "the task gets done" or "exactly one succeeds" reduce to complements and sums of products.

Steps

Step 1: Write down each success and failure probability.

From $P(A)$ and $P(B)$, get $P(A') = 1 - P(A)$ and $P(B') = 1 - P(B)$.

Step 2: "Solved / at least one" — easiest via the complement.

The task fails only if both fail:

(Equivalently $P(A) + P(B) - P(A)P(B)$.) …

Common Mistakes

Mistake 1: Computing "the problem is solved" as $P(A)+P(B)=\tfrac12+\tfrac13=\tfrac56$.

Why it's wrong: that double-counts the case where both solve it. Correct approach: use the complement, $1-P(\text{both fail})=1-\tfrac12\cdot\tfrac23=\tfrac23$ (or subtract $P(\text{both})=\tfrac16$).

Mistake 2: Letting "exactly one solves" include the both-solve case, or dropping one of the two orders. …