Problem 8 - Algebra 2

Question

GARDENING For Exercises 7 and \(8,\) use the following information. Dan is planting 24 irises in his front yard. The flowers he bought were a combination of two varieties, blue and white9. The flowers are not blooming yet, but Dan knows that the probability of having a blue flower is 75\(\% .\) What is the probability that at least 20 of the flowers will be blue?

Step-by-Step Solution

Answer

The probability is approximately 0.2647.

Step 1: Understand the Problem

We need to determine the probability that at least 20 out of 24 irises will be blue, given a 75% chance that each individual iris is blue.

Step 2: Identify the Distribution

Since each flower has a fixed probability of being blue, this is a binomial distribution problem where each flower is a trial. Number of trials is 24 with a success probability of 0.75 (for a blue flower).

Step 3: Define the Parameters

In a binomial distribution, the parameters are: number of trials \(n = 24\), probability of success for each trial \(p = 0.75\), and we need to find the probability of getting at least 20 blue flowers.

Step 4: Use Binomial Probability Formula

The binomial probability formula is \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \(X\) is a random variable representing the number of successes. We will calculate this for \(k\) ranging from 20 to 24.

Step 5: Calculate Individual Probabilities

We calculate the probabilities for \( P(X = 20), P(X = 21), P(X = 22), P(X = 23), \text{ and } P(X = 24)\) and sum them. For instance, \( P(X = 20) = \binom{24}{20} (0.75)^{20} (0.25)^4 \).

Step 6: Use a Calculator or Statistical Table

Since manual calculation involves large numbers, use a calculator or binomial probability tables to find the cumulative probability for \(X \geq 20\).

Step 7: Compute Cumulative Probability

Sum up the individual probabilities computed in Step 5. This will give us the cumulative probability of having at least 20 blue flowers.

Key Concepts

Probability Calculation

In the context of our gardening problem, probability calculation helps us predict the likelihood of specific outcomes. Here, Dan wants to know the chances of having at least 20 blue irises.
Probability is a measure of how likely an event is to occur, represented as a number between 0 and 1, where 0 means the event will not happen at all, and 1 means the event is certain to happen.

• A probability of 0.75 indicates there is a 75% chance that a flower will be blue.
• For our calculation, we assess multiple outcomes: exactly 20, 21, 22, 23, or 24 blue flowers.

Understanding basic probability calculations is crucial because all steps in determining Dan's problem start with assessing each scenario using these equations.

Binomial Probability Formula

The binomial probability formula is a fundamental aspect of this problem. It's used when there are fixed numbers of trials, each with two potential outcomes (success or failure), like Dan’s irises being blue or not.
The formula is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where:

• \( n \) is the total number of trials (24 irises).
• \( k \) is the number of successes (blue flowers) we want.
• \( p \) is the probability of success in each trial (0.75 for blue flowers).
• \( 1-p \) is the probability of failure (0.25 for not blue flowers).

This formula calculates the probability of exactly \( k \) successes. In practice, you would apply this to compute probabilities for each value from 20 to 24 and add those probabilities together for the total probability of success.

Cumulative Probability

Cumulative probability gives us an aggregate view of multiple outcomes. In Dan's situation, he wants to compute the probability of having at least 20 blue flowers.
To achieve this, we calculate the cumulative probability by summing probabilities of all favorable outcomes, i.e., having exactly 20, 21, 22, 23, or 24 blue flowers.

• First, calculate the individual probabilities \( P(X = 20), P(X = 21), ... , P(X = 24) \).
• Next, add all these probabilities together to get the cumulative probability.

This method provides the total probability of at least one outcome happening within the specified range, which, for Dan, means finding the chance of having at least 20 irises that bloom blue.

Binomial Coefficient

The binomial coefficient is a key part of the binomial probability formula. It is written as \( \binom{n}{k} \) and often read as "n choose k."
In simpler terms, it calculates the number of ways to choose \( k \) successes (blue irises) out of \( n \) trials (total irises).

• The mathematical expression for the binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
• For this exercise, that means for each scenario (20,21,22, etc.), you determine how many ways you could pick those specific numbers of blue flowers from 24.

The binomial coefficient helps bridge the conceptual leap from knowing the number of attempts to understanding the combination of successful outcomes achievable in those attempts.