Problem 56 - Algebra and Trigonometry

Question

Probability of Hitting a Target 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 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 are given that the probability of hitting the target is \( p = 0.9 \) and the probability of missing the target is \( q = 0.1 \). We need to find the probability that the archer hits the target exactly 3 times in 5 attempts, using the binomial probability formula.

Step 2: Recall the Binomial Probability Formula

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

Step 3: Calculate the Binomial Coefficient

The binomial coefficient \( \binom{n}{r} \) can be calculated as \( \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \).

Step 4: Substitute Values into the Binomial Formula

Substitute \( n = 5 \), \( r = 3 \), \( p = 0.9 \), and \( q = 0.1 \) into the binomial formula: \[ P(X = 3) = \binom{5}{3} (0.9)^3 (0.1)^2 \].

Step 5: Calculate Components of the Formula

Calculate each part: - \( (0.9)^3 = 0.729 \)- \( (0.1)^2 = 0.01 \)- Therefore, \( 10 \cdot 0.729 \cdot 0.01 = 0.0729 \).

Step 6: Present the Final Answer

The probability that the archer hits the target exactly 3 times in 5 attempts is \( P(X = 3) = 0.0729 \).

Key Concepts

Binomial Distribution

In probability theory, the binomial distribution is a fascinating concept that helps us understand how likely it is for a certain number of successful outcomes to occur in a fixed number of trials. It is largely characterized by two parameters: the number of trials, denoted as \( n \), and the probability of success in each trial, denoted as \( p \).
The binomial distribution is especially useful in situations where each trial is identical and independent. This means that the outcome of one trial doesn't affect the others. For example, each shot of an archer aiming at a target is independent of the other shots.

• The formula for the probability of achieving exactly \( r \) successful outcomes (like hitting a target) in \( n \) trials is \( P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \).
• This can be expanded into situations like flipping a coin, where we identify heads or tails as the success.

Understanding the binomial distribution involves recognizing these independent successes over multiple identical trials, offering insights into real-world phenomena.

Probability Theory

Probability theory is the branch of mathematics that deals with uncertainty. It provides the tools to quantitatively describe and predict the occurrence of events, which are outcomes of interest. At its core, probability theory involves calculating the likelihood of event occurrences.

• Probability values range from 0 to 1, where 0 means the event will never happen, and 1 means it will definitely happen.
• In problems like the one involving the archer, probability theory helps us compute the chance of hitting the target a certain number of times using known probabilities \( p \) and \( q = 1-p \).
• Combining probabilities requires careful consideration of the events' independence or dependence on each other.

Probability theory serves as the foundation for statistics, risk assessment, and decision-making processes in uncertain situations. It's a versatile tool for disciplines ranging from natural sciences to finance.

Binomial Coefficient

A binomial coefficient is a mathematical term used frequently in binomial expansions that represents the number of ways to choose \( r \) successes out of \( n \) trials. It is denoted by \( \binom{n}{r} \) and calculated using the formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( ! \) (factorial) indicates the product of all positive integers up to that number.

• The binomial coefficient shows up in the center of the binomial probability formula: \( P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \).
• Calculating this coefficient correctly can change the probability result significantly, hence why the computation step—\( \binom{5}{3} = 10 \) in our example—is crucial.
• These coefficients also appear in Pascal's Triangle, which is a helpful tool for quick reference in small values of \( n \) and \( r \).

Mastering binomial coefficients aids in solving complex probability problems and is a key part of understanding the broader concept of binomial distribution.