Problem 20 - Algebra 2 and Trigonometry

Question

A landscape company will replace any shrub that they plant if it fails to grow. They estimate that the probability of failure to grow is \(.02 .\) What is the probability that, of the 200 shrubs planted this week, at most 3 must be replaced?

Step-by-Step Solution

Answer

The probability that at most 3 shrubs need to be replaced is approximately 0.857.

Step 1: Identify the Problem Type

The problem involves finding the probability of a certain number of successes (or failures) in a sequence of independent trials. This is a typical scenario for using the binomial distribution since each shrub growing or not growing can be considered a Bernoulli trial.

Step 2: Define the Parameters of the Binomial Distribution

Identify the number of trials, which is the number of shrubs, as \( n = 200 \). The probability of failure (a shrub needing replacement) is \( p = 0.02 \). We need to find the probability of at most 3 failures, which means we want \( P(X \leq 3) \).

Step 3: Use the Binomial Probability Formula

The binomial probability formula is \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \). We'll calculate \( P(X = 0) \), \( P(X = 1) \), \( P(X = 2) \), and \( P(X = 3) \), then sum them to find \( P(X \leq 3) \).

Step 4: Calculate the Individual Probabilities

Compute:\( P(X = 0) = \binom{200}{0} (0.02)^0 (0.98)^{200} \)\( P(X = 1) = \binom{200}{1} (0.02)^1 (0.98)^{199} \)\( P(X = 2) = \binom{200}{2} (0.02)^2 (0.98)^{198} \)\( P(X = 3) = \binom{200}{3} (0.02)^3 (0.98)^{197} \)

Step 5: Compute the Cumulative Probability

Add the probabilities from the previous step:\( P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \). Calculate numerical values from the expressions and sum them up to find the required probability.

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the analysis of random events. It is the foundational framework for understanding and calculating the likelihood of various outcomes. In the context of this exercise, the event of interest is the failure of shrubs to grow, with a failure probability of \(0.02\). Probability theory helps us use this information to find the likelihood of different numbers of shrubs failing to grow. One key aspect of probability theory is that it provides tools like the binomial distribution to model such situations where outcomes are binary—like a shrub either growing or not.

Bernoulli Trials

A Bernoulli trial is a random experiment where there are exactly two possible outcomes: success or failure. In our scenario, each shrub planted represents a Bernoulli trial with:

• "Success" being the shrub grows.
• "Failure" being the shrub does not grow and needs replacement.

Bernoulli trials are essential components of the binomial distribution, as they define the individual experiments that make up the distribution. Since the probability of failure is given as \(0.02\), each shrub's growth scenario is independent of others, all with the same probability of success or failure. This independence is a critical characteristic in analyzing binomial distributions.

Cumulative Probability

Cumulative probability refers to the total probability that the variable takes on a value less than or equal to a specific value. In binomial distributions, it's about finding the sum of probabilities for different outcomes. Here, to find the probability of at most 3 failures, we calculate:

• \( P(X = 0) \)
• \( P(X = 1) \)
• \( P(X = 2) \)
• \( P(X = 3) \)

and add them together to find \( P(X \leq 3) \). This sum gives the cumulative probability of up to 3 shrubs failing, encompassing all possible "successful" scenarios when keeping the failure count low.

Probability of Success

The probability of success in a Bernoulli trial is simply the probability that the desired outcome will occur. In this problem, the probability of a shrub growing (i.e., not failing) is termed the "probability of success" and is given by \(1 - 0.02 = 0.98\). This value, 0.98, reflects the likelihood of each individual shrub successfully growing without needing replacement.
Understanding the probability of success is crucial because it also factors into the probability calculations used in the binomial distribution formula. In the computations for cumulative probability, it's shown as \((1-p)^{n-k}\), where \(n\) is the number of trials and \(k\) is the number of failures. This illustrates the importance of having a firm grasp on both probability of success and failure when using binomial probability.