Problem 11 - Algebra 2 and Trigonometry

Question

In \(7-14,\) answers can be rounded to four decimal places. A store estimates that 1 out of every 25 customers is returning or exchanging merchandise. In the last hour, a cashier had 5 customers. What is the probability that exactly 2 of those were making returns or exchanges?

Step-by-Step Solution

Answer

The probability that exactly 2 of those 5 customers made returns or exchanges is approximately 0.0142.

Step 1: Identify the Problem Type

This is a binomial probability problem. We need to find the probability of a certain number of successes (returns or exchanges) in a fixed number of trials (5 customers).

Step 2: Define the Parameters

We have a binomial distribution with parameters: number of trials \(n = 5\) and probability of success on a single trial \(p = \frac{1}{25} = 0.04\). We need the probability of exactly \(k = 2\) successes.

Step 3: Calculate the Probability Formula

Use the binomial probability formula: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]where \(\binom{n}{k}\) is the binomial coefficient.

Step 4: Compute the Binomial Coefficient

Calculate the binomial coefficient:\[\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10\].

Step 5: Substitute Values Into the Formula

Now, substitute the values into the formula:\[P(X = 2) = 10 \times 0.04^2 \times 0.96^3\].

Step 6: Calculate the Probability

Compute the probability:1. Calculate \(0.04^2 = 0.0016\).2. Calculate \(0.96^3 = 0.884736\).3. Multiply the results: \[10 \times 0.0016 \times 0.884736 = 0.014156416\].

Step 7: Round the Probability

Round the result to four decimal places: \[0.0142\].

Key Concepts

Binomial Distribution

In probability theory, the binomial distribution is a discrete probability distribution. It models the number of successes in a fixed number of trials, with each trial having two possible outcomes: success or failure. **Key Characteristics**:- **Fixed Number of Trials (n)**: The number of customers in this scenario.- **Constant Probability (p)**: Each customer's probability of making a return or exchange. For this problem, it's set at \( p = 0.04 \) or 4%.- **Independence**: Each customer's decision is independent of the others.
This distribution helps us answer questions like "What is the probability of observing a specific number of successes?"
For example, finding the probability of exactly 2 customers returning items out of 5.

Probability of Success

The probability of success in a binomial distribution is critical to solving our problem. It represents the likelihood of one specific outcome occurring in a single trial.**Understanding Probability of Success (p)**:- **Definition**: In this context, it's the chance of a customer returning or exchanging merchandise, given as \( p = \frac{1}{25} = 0.04 \).- **Role in Binomial Distribution**: It's used to calculate the probability of various outcomes over multiple trials.
**Application in Problems**:By knowing the probability of success, you can predict other scenarios, such as no customers making returns or all of them doing so. Calculating these probabilities involves using the formula often and applying the concept effectively.

Binomial Coefficient

The binomial coefficient is a vital component of the binomial probability formula. It counts the number of ways to choose a given number of successes in a fixed trial number.**Calculating the Binomial Coefficient**:- **Formula**: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)- **Example**: For our problem with 5 trials and 2 successes: \( \binom{5}{2} = \frac{5!}{2!(5-2)!} \)
**Why It Matters**:- Determines the number of potential combinations where a specific number of successes can occur.- Essential for accurately calculating the probabilities in a binomial distribution.
The coefficient simplifies complex probability calculations by laying out the foundation of possible outcome combinations.