A box contains 8 white and 3 red balls. If two balls are drawn at random, find the probability that one is white and one is red.

The correct answer is - 24/55

Key Points

• Understanding the Problem
  • The box contains 8 white balls and 3 red balls, for a total of 11 balls.
  • We need to calculate the probability that one white ball and one red ball are drawn when two balls are selected randomly.
• Number of Favorable Outcomes
  • The number of ways to choose one white ball from 8 white balls is given by C(8, 1), which equals 8.
  • The number of ways to choose one red ball from 3 red balls is given by C(3, 1), which equals 3.
  • The total number of favorable outcomes is C(8, 1) × C(3, 1), i.e., 8 × 3 = 24.
• Total Number of Possible Outcomes
  • The total number of ways to choose any two balls from 11 balls is given by C(11, 2).
  • Using the formula for combinations, C(n, r) = n! / [r!(n - r)!], we calculate:
    • C(11, 2) = 11 × 10 / 2 = 55
• Final Probability Calculation
  • Probability = (Number of favorable outcomes) / (Total number of possible outcomes).
  • Hence, the probability = 24 / 55.

Additional Information

• Combinations
  • The formula for combinations is C(n, r) = n! / [r!(n - r)!].
  • It represents the number of ways to choose r items from a total of n items, without considering the order.
  • For example:
    • C(8, 1) means choosing 1 item from 8 items, which equals 8.
    • C(11, 2) means choosing 2 items from 11 items, which equals 55.
• Probability
  • Probability is defined as the ratio of favorable outcomes to total possible outcomes.
  • Formula: P(Event) = (Favorable Outcomes) / (Total Outcomes).
  • It always lies between 0 and 1.
• Key Assumptions in Random Selection
  • The selection is done without replacement, meaning once a ball is drawn, it is not returned to the box.
  • Each ball has an equal probability of being selected.