Understanding Probability of Independent Events
Probability theory is a fundamental branch of mathematics used to quantify uncertainty. One of the most important concepts in probability is the idea of independent events.
What are Independent Events?
Two events are considered independent if the occurrence of one does not affect the probability of the occurrence of the other. In our example, whether person A solves a problem has no bearing on whether person B solves it.
The Multiplication Rule
For any two independent events $A$ and $B$, the probability that both occur simultaneously (denoted as $P(A \cap B)$) is the product of their individual probabilities:
$$P(A \cap B) = P(A) \times P(B)$$
Step-by-Step Solution
Given the problem:
- Probability that A solves the problem: $P(A) = \frac{2}{3}$
- Probability that B solves the problem: $P(B) = \frac{1}{3}$
We are looking for the probability that the problem is solved by both A and B.
- Identify the relationship: Since A and B working on the problem are independent, we use the multiplication rule.
- Apply the formula: $P(A \text{ and } B) = P(A) \times P(B)$
- Substitute the values: $P(A \text{ and } B) = \frac{2}{3} \times \frac{1}{3}$
- Calculate the result: $P(A \text{ and } B) = \frac{2 \times 1}{3 \times 3} = \frac{2}{9}$
Intuition
Think of it as narrowing down the sample space. Since A only succeeds 2 out of 3 times, and of those times, B only succeeds 1 out of 3 times, you are taking a fraction of a fraction. $\frac{1}{3}$ of $\frac{2}{3}$ is $\frac{2}{9}$.