Probability of Drawing Two Strawberries from a Bag
Calculating probabilities in scenarios such as drawing fruits from a bag can be both interesting and insightful. Let's explore the probability of drawing two strawberries from a bag containing 4 strawberries and 8 grapes. This article will break down the process using combinations, a fundamental concept in probability.
Understanding the Problem
A bag contains 4 strawberries and 8 grapes. The goal is to find the probability that both fruits drawn from it are strawberries. Let's begin by defining the elements and their counts.
Total Strawberries: 4 Total Grapes: 8 Total Fruits: 12 (4 8)Calculating the Total Number of Ways to Choose 2 Fruits
The total number of ways to choose 2 fruits out of 12 can be calculated using combinations. Combinations are used when the order of selection does not matter. The formula for combinations is (binom{n}{r} frac{n!}{r!(n-r)!}), where (n) is the total number of items, and (r) is the number of items to choose.
Using this formula:
(binom{12}{2} frac{12!}{2!(12-2)!} frac{12!}{2!10!} frac{12 times 11}{2 times 1} 66)
Calculating the Number of Favorable Outcomes
The number of ways to choose 2 strawberries from 4 can also be calculated using combinations:
(binom{4}{2} frac{4!}{2!(4-2)!} frac{4!}{2!2!} frac{4 times 3}{2 times 1} 6)
Probability Calculation
The probability of drawing 2 strawberries is given by the ratio of the number of favorable outcomes to the total number of outcomes:
(P(text{2 strawberries}) frac{text{Ways to choose 2 strawberries}}{text{Total ways to choose 2 fruits}} frac{6}{66} frac{1}{11})
Conclusion
The probability that both fruits drawn are strawberries is (frac{1}{11}) or approximately 0.0909. This example showcases the application of combinations in solving probability problems, providing a clear and direct method to solve similar scenarios.
Additional Insights
Understanding the concept of combinations and their application in probability can be extended to a wider range of problems in statistics and data analysis. Whether you're dealing with a small bag of fruits or a complex real-world scenario, the principles remain the same: calculate the total number of possible outcomes and the number of favorable outcomes, and then determine the probability as their ratio.