Calculating Probabilities for Randomly Selecting Chocolates
Let's delve into the delightful world of probability by solving the problem of randomly selecting chocolates from a box. The scenario involves selecting three chocolates from a box containing 10 milk chocolates and 15 dark chocolates. We'll analyze the probability of various outcomes: exactly two dark chocolates, exactly two milk chocolates, and exactly two dark chocolates or exactly two milk chocolates. This will be done using the concept of combinations, a fundamental tool in probability theory.
Understanding the Scenario
The total number of chocolates is 25 (10 milk chocolates 15 dark chocolates). We are interested in the ways to choose 3 chocolates out of 25. This can be calculated using the combination formula:
Cn k n! / (k!(n-k)!)
where n is the total number of chocolates, and k is the number of chocolates to be chosen.
The total number of ways to choose 3 chocolates from 25 is:
C25 3 25! / (3!(25-3)!) 2300
Probability of Selecting Exactly Two Dark Chocolates
To find the probability of selecting exactly 2 dark chocolates and 1 milk chocolate:
Ways to choose 2 dark chocolates from 15 dark chocolates:C15 2 15! / (2!(15-2)!) 105
Ways to choose 1 milk chocolate from 10 milk chocolates:C10 1 10
Total ways to choose 2 dark and 1 milk:
Total ways 105 * 10 1050
Probability:
P2 dark 1 milk 1050 / 2300 105 / 230 21 / 46 ≈ 0.4565
Probability of Selecting Exactly Two Milk Chocolates
To find the probability of selecting exactly 2 milk chocolates and 1 dark chocolate:
Ways to choose 2 milk chocolates from 10 milk chocolates:C10 2 10! / (2!(10-2)!) 45
Ways to choose 1 dark chocolate from 15 dark chocolates:C15 1 15
Total ways to choose 2 milk and 1 dark:
Total ways 45 * 15 675
Probability:
P2 milk 1 dark 675 / 2300 135 / 460 ≈ 0.2935
Probability of Selecting Two Dark Chocolates or Two Milk Chocolates
To find the probability of selecting either 2 dark chocolates or 2 milk chocolates, we sum the probabilities from parts A and B:
Probability of 2 dark chocolates: 21 / 46
Probability of 2 milk chocolates: 27 / 92 (as derived from 135 / 460)
Converting 21 / 46 to a common denominator with 27 / 92:
P2 dark or 2 milk (21 * 10) / 460 27 / 460 210 / 460 27 / 460 337 / 460 69 / 92 ≈ 0.75
Final Answers
Probability of exactly two dark chocolates: 21 / 46 ≈ 0.4565 Probability of exactly two milk chocolates: 27 / 92 ≈ 0.2935 Probability of two dark chocolates or two milk chocolates: 69 / 92 ≈ 0.75This detailed analysis demonstrates the power of combinations in solving real-world probability problems. Whether you're a student, a professor, or a casual observer, understanding these concepts can enhance your analytical skills and provide insights into various probabilistic scenarios.