Problem 21 - Algebra 2

Question

As a maintenance manager, Jackie Thomas is responsible for managing the maintenance of an office building. When entering a room after hours, the probability that she selects the correct key on the first try is \(\frac{1}{5} .\) If she enters 6 rooms in an evening, find each probability. \(P(\text { correct exactly } 4 \text { times })\)

Step-by-Step Solution

Answer

P(correct exactly 4 times) ≈ 0.01536.

Step 1: Identify the Type of Probability Problem

This exercise involves finding the probability of a specific number of successes in multiple independent trials, where each trial has the same probability of success. This is a binomial probability problem.

Step 2: Define Binomial Formula Parameters

For a binomial distribution: \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \). Here, \( n = 6 \) (total trials), \( p = \frac{1}{5} \) (probability of success on each trial), and \( k = 4 \) (desired number of successes).

Step 3: Calculate the Binomial Coefficient

The binomial coefficient is calculated as follows: \( \binom{6}{4} = \frac{6 \times 5}{2 \times 1} = 15 \).

Step 4: Calculate the Probability of Success

The probability of exactly 4 successes is \( p^4 = \left(\frac{1}{5}\right)^4 = \frac{1}{625} \).

Step 5: Calculate the Probability of Failure

The probability of failure (choosing wrongly) is \( 1 - p = \frac{4}{5} \). The probability of failing on 2 trials is \( \left(\frac{4}{5}\right)^2 = \frac{16}{25} \).

Step 6: Calculate the Total Probability Using the Binomial Formula

Substitute the values into the binomial formula: \( P(X = 4) = 15 \times \frac{1}{625} \times \frac{16}{25} = \frac{240}{15625} \).

Step 7: Simplify the Resulting Probability

Convert the calculated probability into a simplified or decimal form, if possible. \( \frac{240}{15625} \approx 0.01536 \). This represents the probability that Jackie selects the correct key exactly 4 times.

Key Concepts

Binomial Distribution

In probability and statistics, a binomial distribution is a discrete probability distribution that summarizes the likelihood of a value occurring within a fixed number of independent trials. Each of these trials results in a binary outcome; there are typically two possible outcomes, often referred to as "success" and "failure". The binomial distribution is defined by two parameters: the total number of trials, denoted by \(n\), and the probability of success on an individual trial, denoted by \(p\).

• For example, if Jackie Thomas has six rooms to enter, these represent the six trials.
• If the probability of selecting the correct key is \(\frac{1}{5}\), this is the probability of success \(p\).

The outcomes across all of these trials are collectively described by the binomial distribution, which helps in calculating probabilities for different numbers of successes. Using a formula specifically designed for binomial distribution, we can compute the probability of getting a certain number of successes in these trials.

Probability of Success

The probability of success is a crucial factor when working with binomial distributions. In any given trial, it is the chance that the desired outcome, typically referred to as a "success," will occur. This probability remains constant across all trials in a binomial distribution.

• In the office building scenario, the success is defined as selecting the correct key. Here, the chance that Jackie picks the right key on her first attempt is \(\frac{1}{5}\).
• This probability value remains the same for all six trials (or rooms) she attempts during her shift.

Understanding the probability of success is essential because it allows us to compute specific probabilities using the binomial formula. It directly influences the shape and likelihoods derived from the binomial distribution for various numbers of successes.

Binomial Coefficient

The binomial coefficient is fundamental to calculating the probability in a binomial distribution. It is denoted by \(\binom{n}{k}\), which represents "n choose k." This mathematical concept calculates the number of different possible combinations of successes \(k\) within \(n\) trials.

• For Jackie attempting to use the correct key in 6 trials, and being successful in 4, the binomial coefficient is calculated as \(\binom{6}{4} = 15\).
• It ensures the correct count of different arrangements of successes across the trials by considering the multitude of ways successes can occur.

Calculating this coefficient is vital as it forms a component of the binomial formula, thereby directly impacting the final probability of achieving a specified number of successes.

Independent Trials

In the context of a binomial distribution, independent trials imply that the outcome of one trial does not influence the others. Each trial is an isolated event, and the probability of success or failure remains unaffected and constant.

• Going back to our example, whenever Jackie enters a new room, the chance of using the correct key remains \(\frac{1}{5}\), regardless of her previous attempts' outcomes.
• This independence ensures that every room she attempts is a fresh trial, uninfluenced by any success or error made with previous rooms.

The assumption of independent trials is critical, as it upholds the properties of a binomial distribution. It ensures that the binomial probability formula can be applied correctly, leading to accurate probability computations.