Question

Consider a binomial random variable with \(n=8\) and \(p=.7 .\) Let \(x\) be the number of successes in the sample. Evaluate the probabilities in Exercises \(2-6 .\) $$ P(x \geq 3) $$

Short answer

Answer: The probability of having at least 3 successes in the given binomial random variable is approximately 0.9913.

Step-by-step solution

Understand the Binomial Probability Formula

A binomial random variable is a discrete random variable with two possible outcomes: success (with probability \(p\)) and failure (with probability \(1-p\)). The binomial probability formula gives the probability of observing \(x\) successes in \(n\) trials, as follows: $$ P(x)= \binom{n}{x} p^x(1-p)^{n-x} $$ where \(\binom{n}{x}\) is the binomial coefficient (also known as "combinations"), which can be computed as: $$ \binom{n}{x} = \frac{n!}{x!(n-x)!} $$

Apply the Binomial Formula for Each X from 3 to 8

We want to compute: $$ P(x \geq 3) = P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6) + P(x = 7) + P(x = 8) $$ Now, we apply the binomial probability formula for each \(x\) from \(3\) to \(8\):

Calculate Individual Probabilities

Using the binomial probability formula, we calculate the individual probabilities for each \(x\): \(P(x=3)= \binom{8}{3} (0.7)^3 (0.3)^5\) \(P(x=4)= \binom{8}{4} (0.7)^4 (0.3)^4\) \(P(x=5)= \binom{8}{5} (0.7)^5 (0.3)^3\) \(P(x=6)= \binom{8}{6} (0.7)^6 (0.3)^2\) \(P(x=7)= \binom{8}{7} (0.7)^7 (0.3)^1\) \(P(x=8)= \binom{8}{8} (0.7)^8 (0.3)^0\)

Add the Probabilities

Finally, we add the individual probabilities to compute the desired probability: $$ P(x \geq 3) = P(x=3) + P(x=4) + P(x=5) + P(x=6) + P(x=7) + P(x=8) $$ After calculating the individual probabilities, we obtain the final probability \(P(x \geq 3) \approx 0.9913\). Thus, the probability of having at least \(3\) successes in the given binomial random variable is roughly \(0.9913\).

Key concepts

Binomial Probability Formula

The binomial probability formula is a cornerstone of understanding how to deal with scenarios involving a sequence of independent trials, where each trial only has two possible outcomes: success or failure. This formula allows us to compute the probability of having a certain number of successes, labeled as x, within a fixed number of trials, or n.

To go further into detail, the formula is represented as follows:
\[ P(x) = \binom{n}{x} p^x(1-p)^{n-x} \]
Here, n signifies the total number of trials, p is the probability of success on any given trial, and (1-p) refers to the probability of failure. The exponent x is the number of successes for which we wish to find the probability, and n-x is the number of failures. What makes this formula incredibly versatile is its ability to flexibly calculate the probabilities for different numbers of successes, whether that be zero, all the trials, or any number in between.

An essential piece of the binomial probability formula is the binomial coefficient, \(\binom{n}{x}\), which calculates the number of ways we can choose x successes from n trials, fundamentally rooted in combinations from combinatorial mathematics. This part of the formula adjusts the likelihood calculation to account for every possible combination where x successes could occur across n trials.

Binomial Coefficient

Understanding the binomial coefficient is crucial for not just grasping the binomial probability formula, but for understanding the bigger picture of probability and combinations. The binomial coefficient is commonly written as \(\binom{n}{x}\), which reads as 'n choose x.' It represents the number of unique ways to select x items out of n without considering the order of selection.

Mathematically, it is defined as:\[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \]
Here, the '!' denotes the factorial, where n factorial, denoted as n!, is the product of all positive integers up to n. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

When dealing with problems involving the binomial coefficient, it's important to recognize how the concept is not just a mere numerical value but a representation of the combination's principle. It allows us to sum up all the possible combinations of successes and failures in an organized manner, which is fundamental in calculating binomial probabilities.

Discrete Random Variable

A discrete random variable is a type of random variable that assumes a countable number of distinct values. Unlike continuous random variables, which can take on any value within an interval, discrete random variables are limited to specific values. This characteristic makes them particularly relevant in the realm of binomial distribution and probability, where the outcomes of trials are distinctly categorized as successes or failures.

In any given scenario, such as flipping a coin or rolling a die, the number of possible results is finite and easily listed. In the case of our exercise problem, the binomial random variable denotes the number of successes in a series of 8 trials, with each trial having two possible outcomes. The random variable can take on values from 0 through 8, which are the total possible successes in this case.

The binomial random variable holds significant importance because every possible outcome of the variable correlates to an exact probability computed by the binomial probability formula. This makes such variables a pivotal concept in fields like statistics and probability theory, as they provide a mathematical framework to predict the likelihood of observable phenomena that are random but have a known range of outcomes.