Problem 16 - Algebra and Trigonometry

Question

Archery An archer hits his target 80\(\%\) of the time. If he shoots seven arrows, what is the probability of each event? (a) He never hits the target. (b) He hits the target each time. (c) He hits the target more than once. (d) He hits the target at least five times.

Step-by-Step Solution

Answer

(a) 0.000128; (b) 0.2097; (c) 0.9943; (d) 0.8554.

Step 1: Understanding the Problem

The problem involves the probability of hitting the target at different frequencies. Since success probability (hitting target) is 80\(\%\), we can assume the probability of hitting the target, \(p\), is 0.8, and missing, \(q\), is 0.2. We are working with 7 shots, so it’s a binomial distribution problem with \(n = 7\). The probability of \(k\) successes out of \(n\) trials is given by the binomial formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \].

Step 2: Calculating the Probability of Never Hitting (0 Hits)

For part (a), we calculate the probability of not hitting the target at all (0 hits).\( P(X = 0) = \binom{7}{0} (0.8)^0 (0.2)^7 \). Simplifying, \( \binom{7}{0} = 1 \) and \( (0.8)^0 = 1 \), so \( P(X = 0) = 1 * 1 * (0.2)^7 = (0.2)^7 \approx 0.000128 \).

Step 3: Calculating the Probability of Hitting Every Time (7 Hits)

For part (b), we find the probability of hitting the target all 7 times.\( P(X = 7) = \binom{7}{7} (0.8)^7 (0.2)^0 \). Here, \( \binom{7}{7} = 1 \), and \( (0.2)^0 = 1 \), so \( P(X = 7) = 1 * (0.8)^7 \approx 0.2097152 \).

Step 4: Calculating the Probability of Hitting More Than Once (2 to 7 Hits)

For part (c), we need \( P(X > 1) = 1 - P(X = 0) - P(X = 1) \). We already know \( P(X = 0) \). Determine \( P(X = 1) \): \( P(X = 1) = \binom{7}{1} (0.8)^1 (0.2)^6 = 7 * 0.8 * (0.2)^6 \approx 0.0055296 \). So, \( P(X > 1) = 1 - 0.000128 - 0.0055296 \approx 0.9943424 \).

Step 5: Calculating the Probability of Hitting At Least Five Times (5 to 7 Hits)

For part (d), calculate the probability of hitting at least 5 times: \( P(X \ge 5) = P(X = 5) + P(X = 6) + P(X = 7) \). Calculate each: \( P(X = 5) = \binom{7}{5} (0.8)^5 (0.2)^2 = 21 * (0.8)^5 * (0.2)^2 \approx 0.27869184 \), \( P(X = 6) = \binom{7}{6} (0.8)^6 (0.2)^1 = 7 * (0.8)^6 * 0.2 \approx 0.3670016 \), and we already found \( P(X = 7) \approx 0.2097152 \). Thus, \( P(X \ge 5) \approx 0.27869184+0.3670016+0.2097152 \approx 0.85540864 \).

Key Concepts

Probability

Probability is a fundamental concept in statistics that measures the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
In the case of a binomial distribution, as seen in the archery problem, we are dealing with repeated experiments or trials.
Each trial has exactly two possible outcomes: success or failure.The probability of success, denoted as \( p \), is 0.8 (or 80\% in the archery example).

• The probability of failure (or missing the target) is denoted as \( q \), calculated as \( 1 - p \). In this problem, \( q = 0.2 \).
• The number of trials, \( n \), is the number of arrows shot, which is 7.

These parameters allow us to use the binomial probability formula to find the likelihood of different outcomes, like hitting the target a specific number of times.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations and permutations. In probability problems, like the one we have here, it helps us determine how many ways certain events can occur.
The binomial coefficient, represented as \( \binom{n}{k} \) in the formula, plays a crucial role. It calculates the number of ways \( k \) successes can occur in \( n \) trials.For example, \( \binom{7}{0} \) is used to find the probability of hitting the target zero times. It calculates to 1 since there's only one way to never hit the target in seven shots: missing every single one.
On the other hand, \( \binom{7}{5} \) calculates the ways to hit the target exactly five times out of seven tries, which is significantly more. This is used to determine the probability of outcomes that involve multiple successful hits.
By understanding how to use these combinations, we can evaluate the probability of different scenarios in binomial distributions.

Success Rate

The success rate in probability refers to the fraction of attempts in which a desired outcome occurs.
In the context of our archery problem, the archer's success rate is 80\%, meaning he is expected to hit the target 80 out of every 100 shots.
This success rate is expressed as the probability \( p \) in binomial calculations.To find the probability of different types of success across multiple trials, we consider:

• The probability of hitting all 7 targets: This is a single scenario with a probability calculated using \( p^7 \).
• The likelihood of hitting the target various numbers of times (e.g., never, at least five times): These are calculated using different values of \( k \) in the binomial formula.

Understanding the success rate and its application helps to predict outcomes in repeated experiments, which is the essence of the binomial distribution applied in this problem.