Calculating the Probability of Drawing 2 Strawberry Chocolates from a Box

In a box of 12 chocolates, 4 are strawberry, 4 are orange, and 4 are nutty. The question is often asked: What is the probability of drawing 2 strawberry chocolates? This article will guide you through a step-by-step method to solve this problem using mathematical concepts.

Understanding the Problem

The question involves the concept of probability and combinations, specifically the scenario of drawing chocolates without replacement. This means that once a chocolate is drawn, it is not put back into the box, which affects the probability of subsequent draws.

Step-by-Step Calculation

Step 1: Calculate the Total Number of Ways to Choose 2 Chocolates from 12

The total number of ways to choose 2 chocolates from 12 can be calculated using the combination formula:

binom{n}{r} frac{n!}{r!(n-r)!}

Here, (n) is the total number of items, which is 12, and (r) is the number of items to choose, which is 2. Plugging these values into the formula gives:

binom{12}{2} frac{12!}{2!(12-2)!} frac{12 times 11}{2 times 1} 66

So, there are 66 different ways to choose 2 chocolates from 12.

Step 2: Calculate the Number of Ways to Choose 2 Strawberry Chocolates from 4

Similarly, we use the combination formula to find the number of ways to choose 2 strawberry chocolates from 4:

binom{4}{2} frac{4!}{2!(4-2)!} frac{4 times 3}{2 times 1} 6

There are 6 ways to choose 2 strawberry chocolates from 4.

Step 3: Calculate the Probability

The probability of drawing 2 strawberry chocolates is the ratio of the number of ways to choose 2 strawberry chocolates to the total number of ways to choose 2 chocolates:

P(2 strawberry) frac{text{Number of ways to choose 2 strawberry}}{text{Total number of ways to choose 2 chocolates}} frac{6}{66} frac{1}{11}

Therefore, the probability of drawing 2 strawberry chocolates is (frac{1}{11}).

Detailed Explanation of Probabilities

Alternatively, you can also calculate the probability by considering the individual probabilities of each draw:

The probability of drawing a strawberry chocolate on the first draw is (frac{4}{12}) or (frac{1}{3}). The probability of drawing another strawberry chocolate on the second draw is (frac{3}{11}) since one strawberry chocolate has already been drawn.

The combined probability of both events happening is the product of these individual probabilities:

P(2 strawberry) frac{4}{12} times frac{3}{11} frac{12}{132} frac{1}{11}

In summary, the probability of drawing 2 strawberry chocolates from a box of 12 chocolates is (frac{1}{11}) or approximately 0.0909, or 9.09%. This approach reinforces the fundamental principles of probability and combinations in a practical real-world scenario.