Problem 4 - 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. All successes

Step-by-Step Solution

Answer

The probability of all successes is 0.16807.

Step 1: Identify the Problem

We need to find the probability of getting all successes in 5 independent trials of a binomial experiment where the probability of success is \(p = 0.7\).

Step 2: Use the Binomial Probability Formula

The probability of exactly \(k\) successes in \(n\) independent Bernoulli trials is given by the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]In this problem, \(n = 5\), \(k = 5\), and \(p = 0.7\).

Step 3: Calculate the Binomial Coefficient

The binomial coefficient \(\binom{n}{k}\) is calculated as: \[ \binom{5}{5} = \frac{5!}{5!(5-5)!} = 1 \]

Step 4: Compute the Probability

Substitute the values into the formula: \[ P(X = 5) = 1 \times (0.7)^5 \times (0.3)^0 \]\[ P(X = 5) = (0.7)^5 \]Calculate \((0.7)^5\): \[ (0.7)^5 = 0.16807 \]

Step 5: Finalize the Calculation

Thus, the probability of having all 5 trials result in success is \(0.16807\).

Key Concepts

Understanding Binomial Experiments

A binomial experiment is a statistical experiment that meets certain conditions. Firstly, it has a fixed number of trials, denoted by \( n \). In this exercise, the number of trials is 5. Secondly, it involves independent trials, which means the outcome of each trial does not affect the others. This characteristic is crucial because it ensures that what happens in one trial does not change the probability of outcomes in subsequent trials.

Each trial can result in one of two outcomes, typically called "success" or "failure". These are mutually exclusive events. For our specific example, a success could be defined based on any criteria that fit the scenario, such as flipping a coin resulting in heads or passing a test.

• Fixed number of trials
• Two possible outcomes per trial
• All trials are independent

Understanding this concept properly enables us to apply the binomial probability formula to discover the likelihood of successes in our trials.

Grasping Independent Trials

Independent trials are a key concept in understanding binomial experiments. In essence, this means that the outcome of one trial does not affect the outcome of another.

For example, if flipping a coin is your trial, getting heads on the first flip does not change the probability of getting heads on the second flip. Each flip is independent of all others. This idea is critical because it allows us to apply the binomial probability model consistently without adjusting for previous outcomes.

• Outcomes do not influence each other
• Maintaining same probability across trials
• Applicable for coin tosses, dice rolls, etc.

Understanding independence helps us accurately assess probability using the formula, knowing each trial is a clean slate.

Explaining the Probability of Success

Probability of success is a fundamental part of calculating binomial probability. It is denoted by \( p \), which represents the chance of achieving a success in any given trial. For instance, if you flip a fair coin, the probability of getting heads (a success) is \( 0.5 \).

In our exercise, \( p = 0.7 \), indicating that there is a 70% chance of success on each of the 5 trials. The probability of failure is denoted by \( q \), where \( q = 1 - p \). This means \( q = 0.3 \) here. Therefore, the binomial probability formula \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\] relies heavily on knowing \( p \) and \( q \) for calculating the probability for a specific number of successes.

• Probability of success (\( p \)) is key for calculations
• Related to real-world likelihood of outcomes
• Calculated via the formula for specific successes

Understanding this allows you to apply the binomial formula to find how likely it is to achieve a number of successes given a probability.