Introduction

When a homeowner plants bulbs selecting at random from a box containing a mix of tulips and daffodils, the probability of a specific combination can be calculated using the hypergeometric distribution. This article explores the process of determining the probability of planting exactly 2 daffodil bulbs and 4 tulip bulbs from a box containing 5 tulip bulbs and 4 daffodil bulbs. We will break down the steps involved and provide clear explanations and calculations.

Step-by-Step Calculation

Let's consider the problem of a homeowner planting 6 bulbs from a mix of 5 tulip bulbs and 4 daffodil bulbs. Our objective is to calculate the probability of planting exactly 2 daffodil bulbs and 4 tulip bulbs.

Step 1: Identifying the Total Number of Bulbs and Desired Selection

The total number of bulbs is 9, consisting of 5 tulip bulbs and 4 daffodil bulbs. The homeowner wants to plant 2 daffodil bulbs and 4 tulip bulbs.

Step 2: Calculating the Number of Ways to Choose the Desired Selection

To determine the number of ways to choose the desired bulbs, we use the combinations formula, denoted as C(n, k) or binom{n}{k}.

Choosing 2 Daffodil Bulbs from 4

The number of ways to choose 2 daffodil bulbs from 4 is:

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

Choosing 4 Tulip Bulbs from 5

The number of ways to choose 4 tulip bulbs from 5 is:

C(5, 4) frac{5!}{4!(5-4)!} frac{5}{1} 5

The total number of ways to choose 2 daffodil and 4 tulip bulbs is:

C(4, 2) times C(5, 4) 6 times 5 30

Step 3: Calculating the Total Number of Ways to Choose Any 6 Bulbs From 9

To find the total number of ways to choose any 6 bulbs out of 9, we again use the combinations formula:

C(9, 6) C(9, 3) frac{9!}{3!(9-3)!} frac{9 times 8 times 7}{3 times 2 times 1} 84

Step 4: Calculating the Probability

The probability of planting 2 daffodil bulbs and 4 tulip bulbs is given by the ratio of the number of successful outcomes to the total outcomes:

P frac{30}{84} frac{5}{14}

Final Answer

The probability that the homeowner planted 2 daffodil bulbs and 4 tulip bulbs is:

frac{5}{14}

Conclusion

By following the steps outlined in this article, we have successfully calculated the probability of a specific combination of bulbs being planted. Understanding and applying the hypergeometric distribution in such scenarios can be crucial in various fields, including statistics, probability theory, and data science.

Keywords

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References

DeGroot, M. H., Schervish, M. J. (2011). Probability and Statistics (4th Edition). Pearson.