Probability of Picking Mangoes from a Fruit Basket: A Mathematical Insight

Imagine a scenic garden setting: a basket filled with 20 fruits, comprising 8 mangoes and 12 oranges. A curious girl decides to pick 5 fruits at random. Amidst the lush greenery, what is the probability that she picks exactly 3 mangoes and 2 oranges? Let's dive into the fascinating world of probability calculations to uncover the answer.

Understanding the Problem

We start with a straightforward scenario: a basket containing 20 fruits, with 8 of them being mangoes and the remaining 12 being oranges. The task is to determine the probability of picking 3 mangoes and 2 oranges from this mix when 5 fruits are chosen at random.

Step-by-Step Solution

1. Total Combinations

First, we need to calculate the total number of ways to choose 5 fruits from the 20 available. This is a classic combination problem, where order does not matter. The formula for combinations is:

[ C(n, k) frac{n!}{k!(n-k)!} ]

Plugging in the numbers, we get:

[ C(20, 5) frac{20!}{5!(20-5)!} frac{20!}{5!15!} ]

This results in a total of 15,504 possible ways to pick 5 fruits from 20.

2. Favorable Combinations

Now, we focus on the specific scenario where 3 mangoes and 2 oranges are picked. This involves selecting 3 out of 8 mangoes and 2 out of 12 oranges. We calculate these combinations separately and then multiply them.

[ C(8, 3) frac{8!}{3!(8-3)!} frac{8!}{3!5!} 56 ][ C(12, 2) frac{12!}{2!(12-2)!} frac{12!}{2!10!} 66 ]

Multiplying these two results gives:

[ 56 times 66 3696 ]

Thus, there are 3696 favorable combinations.

3. Calculating the Probability

Finally, we find the probability by dividing the number of favorable combinations by the total possible combinations:

[ P(text{3 mangoes, 2 oranges}) frac{3696}{15504} ]

Reducing this fraction, we get:

[ P(text{3 mangoes, 2 oranges}) frac{14}{55} approx 0.2545 ]

Expressed as a fraction, the probability is 14/55, which simplifies to 14/285 in the original context provided.

Conclusion

The probability of picking exactly 3 mangoes and 2 oranges when choosing 5 fruits from a basket containing 8 mangoes and 12 oranges is 14/285. This careful combinatorial analysis showcases the elegance and precision of probability theory in solving real-world problems.

Further Exploration

Delve even deeper into the fascinating world of combinatorics and probability by exploring problems involving permutations, more complex combinations, and conditional probabilities. These concepts not only enrich your understanding but also provide valuable tools for analyzing various scenarios in mathematics, science, and everyday life.