Calculating the Probability of Drawing Exactly 3 Walnuts from a Bag of Mixed Nuts
In this article, we will explore the mathematical principles behind calculating the probability of drawing a specific combination of nuts from a bag containing a mixed variety. Specifically, we will focus on the scenario where a bag contains 9 hazelnuts, 10 almonds, and 5 walnuts. We will determine the probability of drawing exactly 3 walnuts when 6 nuts are randomly drawn from the bag.
The Problem Statement
Our scenario involves a bag containing a total of 24 nuts, composed of 9 hazelnuts, 10 almonds, and 5 walnuts. If 6 nuts are drawn at random, what is the probability of drawing exactly 3 walnuts?
Understanding the Combinatorial Problem
The first step in solving this problem involves determining the total number of ways to draw 6 nuts from the bag of 24 nuts. This can be calculated using the combination formula, which is given by:
[ {24 choose 6} frac{24!}{6!(24-6)!} 134596 ]
Next, we need to determine the number of ways to draw exactly 3 walnuts from the 5 walnuts available. This is also a combinatorial problem, which can be expressed as:
[ {5 choose 3} frac{5!}{3!(5-3)!} 10 ]
The remaining 3 nuts must come from the 19 non-walnut nuts (9 hazelnuts and 10 almonds). The number of ways to choose 3 nuts from these 19 is:
[ {19 choose 3} frac{19!}{3!(19-3)!} 969 ]
Therefore, the total number of favorable outcomes (drawing exactly 3 walnuts and 3 non-walnut nuts) is given by:
[ {5 choose 3} times {19 choose 3} 10 times 969 9690 ]
The Probability Calculation
The probability of drawing exactly 3 walnuts when 6 nuts are drawn at random from the bag is the ratio of the number of favorable outcomes to the total number of possible outcomes. This can be expressed as:
[ P frac{{5 choose 3} times {19 choose 3}}{{24 choose 6}} frac{9690}{134596} approx 0.0721 ]
This can be simplified to a fraction:
[ P frac{9690}{134596} frac{5145}{67298} approx 0.0761 ]
Alternative Methods of Calculation
There are several other ways to tackle this problem, depending on the specific constraints and the approach one chooses. For instance, one might consider drawing 6 nuts in such a way that there are an equal number of each type of nut. However, the problem statement specifically asks for the exact probability of drawing exactly 3 walnuts.
Conclusion
Understanding the combinatorial principles and probability is crucial for solving such problems. The probability of drawing exactly 3 walnuts when 6 nuts are drawn at random from a bag containing 9 hazelnuts, 10 almonds, and 5 walnuts is approximately 0.0721 or ( frac{9690}{134596} ).