Problem 3 - Algebra and Trigonometry

Question

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. Exactly two successes

Step-by-Step Solution

Answer

The probability of exactly two successes is 0.1323.

Step 1: Understand the Binomial Distribution Formula

The probability of obtaining exactly "k" successes in "n" independent binomial trials is given by the formula: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]where \(\binom{n}{k}\) is the binomial coefficient, \(p\) is the probability of success, and \(1-p\) is the probability of failure.

Step 2: Identify Variables

In this problem, the number of trials \(n = 5\), the number of successes we want \(k = 2\), the probability of success \(p = 0.7\), and the probability of failure is \(q = 0.3\).

Step 3: Calculate the Binomial Coefficient

The binomial coefficient \(\binom{5}{2}\) represents the number of ways to choose 2 successes out of 5 trials. This is calculated as:\[\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10\]

Step 4: Plug Values into the Binomial Formula

Using the binomial formula:\[P(X = 2) = \binom{5}{2} p^2 (1-p)^{5-2}\]Substituting the known values:\[P(X = 2) = 10 \times (0.7)^2 \times (0.3)^3\]

Step 5: Compute the Probability

Calculate the probability by evaluating the expression:\[P(X = 2) = 10 \times 0.49 \times 0.027 = 10 \times 0.01323 = 0.1323\]

Key Concepts

Probability of Success

In any binomial distribution problem, the core element is understanding the probability of success. The probability of success, often represented as \( p \), is the likelihood that a single trial will result in a favorable outcome. In our given exercise, it is stated that \( p = 0.7 \). This means that each trial has a 70% chance of success.

Let's break it down further. The probability of success \( p \) is crucial because it determines how probable it is to achieve a certain number of successes across multiple trials. This forms the basis when calculating the probability of an event in a binomial setting. If you increase the probability of success, you generally increase the likelihood of achieving more successes.

• \( p = 0.7 \) signifies the chance that any single trial is a success.
• \( q = 1 - p = 0.3 \), represents the probability of failure.

These probabilities are foundational blocks when using the binomial formula to calculate the likelihood of experiencing a certain number of successes in multiple trials.

Binomial Coefficient

The binomial coefficient is a key component when working with binomial distributions. It is notated as \( \binom{n}{k} \) and calculates how many combinations or ways we can choose \( k \) successes in \( n \) independent trials.

The formula for the binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n! \) (n factorial) is the product of all positive integers up to \( n \). This helps determine all possible combinations.

• In our example, \( \binom{5}{2} \) is calculated as \( \frac{5 \times 4}{2 \times 1} = 10 \).
• This number signifies there are 10 different ways of achieving exactly 2 successes in 5 trials.

This coefficient acts like a multiplier in the binomial probability formula. It tells us how to appropriately weigh the success probability raised to the number of successes, and the failure probability raised to the remainder of the trials.

Independent Trials

Understanding the concept of independent trials is crucial when dealing with binomial distributions. Independent trials imply that the outcome of one trial does not influence the outcome of another. Each trial in a binomial experiment is separate from the others.

In the context of our problem, we have 5 independent trials. This means that the success or failure of one trial has no bearing on the next. This independence is important because it allows us to multiply the probability of success across trials without any dependencies.

• Each trial's outcome remains unaffected by the preceding trials.
• Each event's probability calculation uses the same probability of success \( p \) intact throughout all trials.

Independent trials are what allow us to use the probability formula for binomial distributions, and without them, calculating exact probabilities would be more complex.