Problem 56 - Precalculus Mathematics for Calculus

Question

The probability that an archer hits the target is \(p=0.9,\) so the probability that he misses the target is \(q=0.1 .\) It is known that in this situation the probability that the archer hits the target exactly \(r\) times in \(n\) attempts is given by the term containing \(p^{r}\) in the binomial expansion of \((p+q)^{n}\). Find the probability that the archer hits the target exactly three times in five attempts.

Step-by-Step Solution

Answer

The probability is 0.0729.

Step 1: Understand the Problem

We have an archer with a probability of hitting the target as \( p = 0.9 \) and missing the target as \( q = 0.1 \). We are asked to find the probability that the archer hits the target exactly "3 times" out of "5 attempts".

Step 2: Use the Binomial Probability Formula

The probability of hitting the target exactly \( r \) times in \( n \) attempts follows a binomial distribution. The binomial probability is given by: \[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]For our problem: - \( n = 5 \) (number of attempts)- \( r = 3 \) (exact number of hits)- \( p = 0.9 \) (probability of hitting the target)- \( q = 0.1 \) (probability of missing the target)

Step 3: Calculate the Binomial Coefficient

Compute the binomial coefficient \( \binom{5}{3} \), which represents "5 choose 3" or the number of ways to choose 3 hits out of 5 attempts:\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

Step 4: Compute the Powers of Success and Failure

Calculate the power of success, \( p^3 \):\[ p^3 = (0.9)^3 = 0.729 \]Calculate the power of failure, \( q^{5-3} = q^2 \):\[ q^2 = (0.1)^2 = 0.01 \]

Step 5: Calculate the Probability

Substitute the values into the binomial formula to find the probability of exactly 3 hits:\[ P(X = 3) = \binom{5}{3} p^3 q^2 = 10 \times 0.729 \times 0.01 = 0.0729 \]

Step 6: State the Final Probability

Thus, the probability that the archer hits the target exactly three times in five attempts is 0.0729.

Key Concepts

Understanding Probability Theory

Probability theory is a branch of mathematics that deals with calculating how likely an event is to occur. It's a foundational concept for statistics and important for making predictions about future events. In this context, we focus on discrete probability, which deals with events that occur in countable sample spaces. An example is the probability of an archer hitting a target, as in our task.

Key aspects of probability theory include:

• Events: Outcomes or sets of outcomes from a random experiment. For the archer, each shot is an event.
• Probability: A numerical value between 0 and 1 that quantifies how likely an event is to happen. A probability of 0 means impossibility, while 1 means certainty.
• Complementary Events: If hitting the target is an event, missing it is the complementary event.

Probability problems often require a structured approach, particularly when multiple events occur. That's where using specific formulas, like the binomial probability formula, can help.

Exploring the Binomial Coefficient

The binomial coefficient is a crucial part of solving problems involving binomial distributions. It is represented as \( \binom{n}{r} \) and is read as "n choose r". Its primary function is to calculate the number of ways to choose \( r \) successes (or hits, in our case) out of \( n \) total trials. Here's how it works:

- The formula is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where "!" denotes a factorial.

• Factorial (\( n! \)): Means multiplying all whole numbers from 1 up to \( n \).

- For our archer scenario, we calculated \( \binom{5}{3} \):

• This means choosing 3 successful hits from 5 total shots.
• Using the formula: \( \frac{5!}{3! \times 2!} = 10 \).
• This tells us there are 10 ways the archer can hit the target exactly 3 times.

Understanding the binomial coefficient helps us tackle problems where each trial has two possible outcomes, such as success or failure.

APPLICATION OF THE BINOMIAL PROBABILITY FORMULA

The binomial probability formula allows us to calculate the probability of achieving exactly \( r \) successes in \( n \) independent trials, with each trial having the same probability of success \( p \). The formula is given by: \[ P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \] Here, each component plays a role:

- \( \binom{n}{r} \): Indicates the possible ways to achieve \( r \) successes.- \( p^r \): Represents the probability of \( r \) successes happening.- \( (1-p)^{n-r} \): Stands for the probability of the other \( n-r \) trials resulting in failure.For the archer hitting the target 3 times in 5 attempts with \( p = 0.9 \):

• Calculate \( \binom{5}{3} = 10 \)
• Power of success: \( 0.9^3 = 0.729 \)
• Power of failure: \( 0.1^2 = 0.01 \)

The probability becomes \( 10 \times 0.729 \times 0.01 = 0.0729 \). This result directly tells us the chance of the archer hitting the target exactly three times. The binomial probability formula simplifies complex event calculations by combining mathematical precision with logical sequence.