Problem 36 - Algebra and Trigonometry

Question

Hitting a Target An archer normally hits the target with probability of \(0.6 .\) She hires a new coach for a series of special lessons. After the lessons she hits the target in five out of eight attempts. (a) Find the probability that she would have hit five or more out of the eight attempts before her lessons with the new coach. (b) Did the new coaching effective if the probability in part (a) is 0.05 or less.)

Step-by-Step Solution

Answer

(a) Calculate P(X>=5) using the binomial formula. (b) Compare P(X>=5) with 0.05.

Step 1: Understanding the Problem

We need to calculate the probability of the archer hitting the target 5 or more times out of 8 attempts with an initial probability of hitting the target at 0.6.

Step 2: Choosing the Probability Distribution

Since we're dealing with a fixed number of independent attempts, each with the same probability of success, we use the binomial distribution formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \( n \) is the number of trials, \( k \) is the number of successful outcomes, and \( p \) is the probability of success on a single trial.

Step 3: Calculating for 5 Hits

Calculate the probability of hitting exactly 5 times out of 8 attempts: \[ P(X = 5) = \binom{8}{5} (0.6)^5 (0.4)^3 \]

Step 4: Calculating for 6 Hits

Calculate the probability of hitting exactly 6 times: \[ P(X = 6) = \binom{8}{6} (0.6)^6 (0.4)^2 \]

Step 5: Calculating for 7 Hits

Calculate the probability of hitting exactly 7 times: \[ P(X = 7) = \binom{8}{7} (0.6)^7 (0.4)^1 \]

Step 6: Calculating for 8 Hits

Calculate the probability of hitting exactly 8 times: \[ P(X = 8) = \binom{8}{8} (0.6)^8 (0.4)^0 \]

Step 7: Summing Probabilities for 5 or More Hits

The total probability of the archer hitting the target five or more times is the sum of probabilities from 5 to 8 hits: \[ P(X \geq 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) \]

Step 8: Comparison with Significance Level

If the probability calculated in the previous step is less than or equal to 0.05, we can conclude that the new coaching was effective.

Key Concepts

Understanding Probability

Probability is a fundamental concept in statistics that quantifies the likelihood of a specific event occurring. It is expressed as a number between 0 and 1. A probability of 0 means the event will not occur, and a probability of 1 means it is certain to occur.

In the context of the archer's performance, the probability of hitting the target initially is 0.6 (60%). This indicates a moderate level of certainty that she will hit the target on any given attempt before her coaching.

It's crucial to understand that while individual attempts might not always result in a hit or miss based precisely on this probability, over a large number of trials, the outcomes will approximate this probability more closely. This foundational understanding helps in making sense of models like the binomial distribution.

The Concept of Independent Trials

Independent trials refer to situations where the outcome of one trial does not affect the outcomes of subsequent trials. Each trial is self-contained, and previous results do not influence future ones. This concept is central to many statistical methods, including the binomial distribution.

In the exercise of hitting targets, each shot by the archer is considered an independent trial. The outcome of hitting or missing the target in one attempt doesn’t dictate what will happen in the next attempt. This independence is crucial for correctly applying the binomial probability formula.

Understanding this independence helps in setting the stage for calculating the total probability of various outcomes across multiple attempts, like determining how often an event might occur out of a set number of trials.

Assessing Probability of Success

The probability of success refers to the chance of the desired outcome happening in a single trial. In our scenario, hitting the target is considered a success, and the probability of success initially is 0.6.

This value is critical when applying statistical models to predict or analyze outcomes. After the coaching, we want to see if the probability of success has changed by assessing real outcomes compared to what was initially expected.

Monitoring the probability of success can indicate whether interventions—such as coaching in this case—are effective or not. By comparing results from before and after the coaching, and understanding the role of random variation in outcomes, we can make informed judgments about performance changes.

Understanding the Binomial Probability Formula

The binomial probability formula is a key tool in statistics used to calculate the probability of a certain number of successes in a fixed number of independent trials. It is expressed as: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where:

• \( n \) is the total number of trials,
• \( k \) is the number of successful outcomes,
• \( p \) is the probability of success on an individual trial,
• \( 1-p \) represents the probability of failure.

In our archer example, we used this formula to determine the probability of hitting the target five or more times out of eight attempts. By calculating probabilities for each potential success count from 5 to 8 and adding them together, we find the cumulative probability for 5 or more successes.

Understanding and using this formula allows for precise probability calculations, helping to evaluate whether interventions or changes, like coaching, lead to statistically significant improvements.