Question

There are 4 white and 2 black balls in a bag and in another bag 3 white and 5 black balls. Find the probability of getting both black balls if a ball is drawn from each bag.

Hint:

In probability, the word "and" for independent events usually means you should multiply the probabilities. The word "or" usually means you should add them (and subtract the intersection if they are not mutually exclusive). Recognizing these keywords can help you quickly set up the problem.

Answer 1

Step 1: Understanding the Concept:
This problem involves calculating the probability of two independent events occurring together. The outcome of drawing a ball from the first bag does not affect the outcome of drawing a ball from the second bag. For independent events, the probability that both occur is the product of their individual probabilities.
Step 2: Key Formula or Approach:
Let A be the event of drawing a black ball from the first bag, and B be the event of drawing a black ball from the second bag.
The probability of an event is given by: \[ P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \] Since the events are independent, the probability of both events occurring is:
\[ P(A \text{ and } B) = P(A) \times P(B) \] Step 3: Detailed Explanation or Calculation:
First Bag:
Total number of balls = 4 white + 2 black = 6 balls.
Number of black balls = 2.
The probability of drawing a black ball from the first bag is \( P(A) \).
\[ P(A) = \frac{\text{Number of black balls}}{\text{Total number of balls}} = \frac{2}{6} = \frac{1}{3} \] Second Bag:
Total number of balls = 3 white + 5 black = 8 balls.
Number of black balls = 5.
The probability of drawing a black ball from the second bag is \( P(B) \).
\[ P(B) = \frac{\text{Number of black balls}}{\text{Total number of balls}} = \frac{5}{8} \] Combined Probability:
The probability of getting a black ball from both bags is the product of the individual probabilities.
\[ P(\text{both black}) = P(A) \times P(B) = \frac{1}{3} \times \frac{5}{8} = \frac{5}{24} \] Step 4: Final Answer:
The probability of getting both black balls is \( \frac{5}{24} \).