Problem 26 - Algebra 2

Question

Open-Ended Describe a binomial experiment that can be solved using the expression \(7^{\mathrm{C}}_{2}(0.6)^{2}(0.4)^{5} .\)

Step-by-Step Solution

Answer

The given expression describes a binomial experiment in which a basketball player with a 60% free throw success rate shoots 7 times and we are finding the probability of them making exactly 2 baskets.

Step 1: Identify Binomial Experiment

The binomial expression provided can represent any number of experiments. One example could be a basketball player who shoots their free throws with a success rate of 60%. A binomial experiment can be identified as the player shooting the ball 7 times and wanting to know the probability of scoring exactly twice.

Step 2: Define Success and Failure Conditions

In this context, 'success' is scoring a basket, which the player does with a probability of 0.6. Conversely, 'failure' is missing the basket, which happens with a probability of 0.4.

Step 3: Apply Binomial Distribution Formula

The binomial distribution for the probability of the player scoring exactly twice (k) in seven attempts (n), can be represented by the same expression given in the problem: theirs \(7^{\mathrm{C}}_{2}(0.6)^{2}(0.4)^{5}\). This represents the number of combinations of 2 successful shots in 7 attempts, multiplied by the probability of scoring in a single attempt (0.6) to the power of the number of successes, multiplied by the probability of missing (0.4) to the power of the number of failures.

Step 4: Interpret the Result

The value obtained from \(7^{\mathrm{C}}_{2}(0.6)^{2}(0.4)^{5}\) is the probability that the player makes exactly 2 baskets out of the 7 attempts, given their shooting probabilities.

Key Concepts

Binomial Distribution

In probability and statistics, a **binomial distribution** describes the number of successes in a fixed number of independent experiments, each having two possible outcomes: success or failure. The experiments must satisfy four main conditions to be considered binomial:

• The number of trials, denoted as \( n \), is fixed.
• Each trial is independent from one another.
• There are only two possible outcomes for each trial, typically designated as "success" or "failure".
• The probability of success is the same for each trial.

In the context of the example given, determining the number of successful shots a basketball player makes in 7 attempts utilizes a binomial distribution. Here, the fixed number of trials \( n \) is 7, and the probability of making a shot (a success) remains consistently at 0.6 throughout each of these attempts.

Probability of Success

The **probability of success** in any given trial of a binomial experiment is critical to computing the overall probability of a certain number of successes. It is denoted by \( p \), where:

• \( p = 0.6 \) in the basketball example, meaning the player has a 60% chance to make a basket on each attempt.

To find the probability of observing exactly \( k \) successful results (e.g., 2 successful shots) in \( n \) trials, you multiply the probability of success raised to the power of \( k \) by the probability of failure raised to the power of \( (n-k) \). For each success-failure combination, the binomial coefficient, represented here as \( 7^{\mathrm{C}}_{2} \), is essential. This calculates the possible ways to achieve exactly 2 successes in 7 tries.

Combinatorics

**Combinatorics** plays an important role in calculating the outcomes in a binomial distribution. It involves counting, arranging, and grouping different combinations of items or events. In the binomial expression \( 7^{\mathrm{C}}_{2} \), combinatorics helps determine the number of ways to choose 2 successes out of 7 trials.The general formula for the number of combinations (\( n^{\mathrm{C}}_{k} \)) is given by:\[^nC_k = \frac{n!}{k!(n-k)!}\]In this formula:

• \( n! \) is the factorial of all integers up to \( n \).
• \( k! \) is the factorial of all integers up to \( k \).
• \( (n-k)! \) is the factorial of all integers up to \( (n-k) \).

Using this, \( 7^{\mathrm{C}}_{2} \) equals \( \frac{7!}{2!(5!)} \), calculating the number of different ways 2 successful shots can occur in 7 attempts.

Probability Theory

At its core, **probability theory** is the mathematical framework used to analyze random phenomena and predict the likelihood of various outcomes. It underpins the concept of a binomial distribution by providing a systematic way to calculate probabilities. Understanding the probability theory involves:

• Recognizing that the sum of probabilities of all possible outcomes is 1.
• Knowing the probability of mutually exclusive events can be added together.

In a binomial experiment, we use probability theory to deal with discrete random variables to compute probabilities of different scenarios—like hitting exactly 2 baskets out of 7 tries where prior successes and failures are independent and have fixed probabilities of 0.6 and 0.4 respectively. By implementing these rules, we can confidently use a binomial distribution to predict and interpret outcomes in real-life scenarios and experiments.