Problem 8 - 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. At least three successes

Step-by-Step Solution

Answer

The probability of at least three successes is 0.83692.

Step 1: Define the binomial probability formula

For a binomial distribution, the probability of getting exactly \( k \) successes in \( n \) trials is given by the formula: \[ P(X = k) = \binom{n}{k} p^k q^{n-k} \] where \( \binom{n}{k} \) is the binomial coefficient calculated as \( \frac{n!}{k! (n-k)!} \).

Step 2: Calculate the probability of exactly 3 successes

We use the binomial formula for \( k=3 \), \( n=5 \), \( p=0.7 \), and \( q=0.3 \): \[ P(X = 3) = \binom{5}{3} (0.7)^3 (0.3)^2 \] Calculate \( \binom{5}{3} = 10 \), then \[ P(X = 3) = 10 \times (0.7)^3 \times (0.3)^2 = 10 \times 0.343 \times 0.09 = 0.3087 \]

Step 3: Calculate the probability of exactly 4 successes

For \( k=4 \), the probability is:\[ P(X = 4) = \binom{5}{4} (0.7)^4 (0.3)^1 \] Calculate \( \binom{5}{4} = 5 \), then \[ P(X = 4) = 5 \times (0.7)^4 \times 0.3 = 5 \times 0.2401 \times 0.3 = 0.36015 \]

Step 4: Calculate the probability of exactly 5 successes

For \( k=5 \), the probability is:\[ P(X = 5) = \binom{5}{5} (0.7)^5 (0.3)^0 \] Calculate \( \binom{5}{5} = 1 \), then \[ P(X = 5) = 1 \times (0.7)^5 \times 1 = 0.16807 \]

Step 5: Sum probabilities for "at least three" successes

Add the probabilities of having exactly 3, 4, and 5 successes:\[ P(X \geq 3) = P(X=3) + P(X=4) + P(X=5) \]\[ P(X \geq 3) = 0.3087 + 0.36015 + 0.16807 = 0.83692 \]

Key Concepts

Probability of Success

In a binomial distribution, each trial results in one of two possible outcomes. One is defined as a "success" and the other as a "failure." The probability of success, denoted as \( p \), is a crucial element that drives the distribution. If you are performing an experiment or conducting trials, you should always determine what constitutes a success in your specific context. Please note that:

• Success is simply a term used in probability. It does not imply a positive outcome in all contexts.
• The sum of the probabilities for success and failure is always 1. Therefore, if \( p=0.7 \), then \( q=1-p=0.3 \).

This is because every trial must result in either a success or a failure. In this exercise, with a success probability of 0.7, it means when you perform a trial, there's a 70% chance of achieving what you've defined as success.Understanding the probability of success helps in calculating the specific outcomes and expected values in multiple trials.

Binomial Coefficient

A key part of calculating binomial probabilities is the binomial coefficient, represented as \( \binom{n}{k} \). This coefficient determines the number of ways to choose \( k \) successes from \( n \) trials and is a central element of the binomial probability formula.
The coefficient is calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Where \(!\) is the factorial function, and it tells us how to distribute those successes across the number of trials.

• For example, with \( n=5 \) trials and \( k=3 \) successes, \( \binom{5}{3} = 10 \).
• Factorials grow rapidly; \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). So, calculating factorials can involve large numbers.

Understanding binomial coefficients is crucial for grasping how probabilities are allocated across different possible outcomes in a binomial distribution.

Independent Trials

In a binomial experiment, it is assumed that the trials are independent. This means that the outcome of one trial does not affect the outcome of another. Each time you conduct a trial, it resets the conditions so that the previous results do not impact what comes next.

• The probability of success remains constant throughout all trials due to this independence.
• This concept is important for the binomial distribution to hold true. If trials were dependent, conclusions drawn using the binomial model might be inaccurate.

For instance, in this exercise where each trial has a 0.7 probability of success, this probability does not change, no matter the outcomes of previous trials. Embracing the independence of trials allows for predictable and repeatable outcomes when using the binomial model.

Probability of Failure

The probability of failure, denoted by \( q \), is the complement of the probability of success. In simpler terms, it is the probability that a single trial will result in failure, rather than success. Mathematically, you can express it as:\[q = 1 - p\]Where \( p \) is the probability of success. Understanding this is useful because:

• It completes the binary nature of the binomial process - either a success or a failure in each trial.
• The probability of failure is crucial when you calculate binomial probabilities, as indicated in the formula \( p^k \times q^{(n-k)} \).

For example, with a probability of success of 0.7, the probability of failure in each trial becomes 0.3, as calculated by \( q = 1 - 0.7 = 0.3 \).In any analysis or calculation within a binomial framework, the probability of failure needs to be properly considered alongside the probability of success to capture the full dynamics of trials.