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 Binomial Distribution

The problem involves a binomial distribution where the probability of hitting, \( p = 0.9 \), and missing, \( q = 0.1 \). The number of trials, \( n = 5 \), and we need to find the probability of exactly \( r = 3 \) hits.

Step 2: Formula Application

The probability of exactly \( r \) successes in \( n \) trials is given by the binomial probability formula:\[P(X = r) = \binom{n}{r} p^r q^{n-r}\]where \( \binom{n}{r} \) is the binomial coefficient.

Step 3: Calculate the Binomial Coefficient

Calculate the binomial coefficient \( \binom{5}{3} \):\[\binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10\]

Step 4: Calculate the Probability Using the Formula

Using the values in the formula:\[P(X = 3) = 10 \times (0.9)^3 \times (0.1)^{5-3}\]Calculate \( (0.9)^3 \) and \( (0.1)^2 \):\[(0.9)^3 = 0.729 \quad \text{and} \quad (0.1)^2 = 0.01\]Thus,\[P(X = 3) = 10 \times 0.729 \times 0.01 = 0.0729\]

Step 5: Present the Final Answer

The probability that the archer hits the target exactly three times in five attempts is \(0.0729\).

Key Concepts

Understanding the Binomial Probability Formula

The binomial probability formula is crucial in finding the likelihood of a specific number of successes in a sequence of trials, where each trial is independent. This formula expresses the probability of exactly \( r \) successes in \( n \) trials. The representation of the formula is:\[P(X = r) = \binom{n}{r} p^r q^{n-r}\]Here's a breakdown:

• \( P(X = r) \) is the probability of getting \( r \) successes.
• \( \binom{n}{r} \) stands for the binomial coefficient that helps determine the number of ways to choose \( r \) successes from \( n \) trials.
• \( p^r \) indicates the probability of success raised to the power of \( r \).
• \( q^{n-r} \) symbolizes the probability of failure, raised to the power of \( n-r \).

Each of these elements comes together to provide the probability for a specific number of successes in given trials. This formula is widely used in statistics for scenarios involving repeated experiments.

Decoding the Binomial Coefficient

The binomial coefficient \( \binom{n}{r} \) is essential for counting the number of ways to choose \( r \) successes from \( n \) trials or attempts. Mathematically, it is expressed as:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]

• \( n! \) (n factorial) means multiplying all whole numbers from 1 up to \( n \).
• The denominator \( r! \) accounts for arrangements of the \( r \) successes, while \( (n-r)! \) accounts for the \( n-r \) failures.

In practical terms, if you want to calculate \( \binom{5}{3} \), it would be:\[\binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10\]This calculation indicates there are 10 distinct ways to have exactly 3 successes in 5 trials.

Probability Calculation for Binomial Distribution

Once we determine the binomial coefficient, probability calculations follow with substituting known values into the binomial probability formula. Using the example of an archer's trials, where the probability \( p = 0.9 \) of hitting the target, and \( q = 0.1 \) for missing, we find the exact probability of an exact number of hits:First, apply the formula:\[P(X = 3) = 10 \times (0.9)^3 \times (0.1)^{5-3}\]Calculations:

• Calculate \( (0.9)^3 = 0.729 \)
• Then \( (0.1)^2 = 0.01 \)
• Thus, \( P(X = 3) = 10 \times 0.729 \times 0.01 \)which results in \( 0.0729 \)

This means the probability that the archer hits the target exactly three times in five attempts is \( 0.0729 \). This step demonstrates applying theoretical knowledge to practical scenarios, highlighting the importance of understanding each component of the formula.