Question

Let \(x\) be a binomial random variable with \(n=7\) and \(p=.5 .\) Find the values of the quantities in Exercises \(11-15 .\) $$ P(x>1) $$

Short answer

Answer: The probability that the binomial random variable x is greater than 1 is approximately 0.8828.

Step-by-step solution

Define the binomial probability mass function

The probability mass function of a binomial random variable \(x\) with parameters \(n\) and \(p\) is given by: $$ P(x=k)=\binom{n}{k}p^k(1-p)^{(n-k)} $$ where \(k\) is the number of successes, \(\binom{n}{k}\) is the number of combinations of choosing \(k\) items from \(n\), and \(p\) is the probability of success.

Calculate P(x ≤ 1) using the binomial PMF

Since we know that \(x\) can only be an integer value between 0 and \(n\) (inclusive), we will need to find the individual probabilities \(P(x=0)\) and \(P(x=1)\): $$ P(x=0)=\binom{7}{0}(0.5)^0(1-0.5)^{7-0} $$ $$ P(x=1)=\binom{7}{1}(0.5)^1(1-0.5)^{7-1} $$ Calculate these probabilities: $$ P(x=0)=\binom{7}{0}(1)(0.5)^7=1\times0.0078125≈0.0078 $$ $$ P(x=1)=\binom{7}{1}(0.5)(0.5)^6=7\times0.015625≈0.1094 $$

Use the principle of complementarity to find P(x > 1)

The principle of complementarity states that: $$ P(x>1)=1-P(x\leq1) $$ We have calculated \(P(x=0)\) and \(P(x=1)\) in the previous step. Now add these probabilities to find \(P(x≤1)\): $$ P(x\leq1)=P(x=0)+P(x=1)≈0.0078+0.1094≈0.1172 $$ Subtract \(P(x\leq1)\) from 1 to find \(P(x>1)\): $$ P(x>1)=1-P(x\leq1)≈1-0.1172≈0.8828 $$ So, the probability that the binomial random variable \(x\) is greater than 1 is approximately 0.8828.

Key concepts

Binomial Random Variable

Understanding the concept of a binomial random variable is crucial to navigating problems involving discrete probability distributions. A binomial random variable is a specific type of random variable that has only two possible outcomes, typically labeled as 'success' and 'failure' for each trial. For example, flipping a coin results in either heads or tails, which can be coded as success or failure respectively.

The binomial random variable is characterized by the number of trials, denoted as 'n', and the probability of success, 'p', in each trial. The trials are assumed to be independent, meaning the outcome of one trial does not affect the outcome of another. In the exercise provided, we explore the case where the number of trials is 7, and the probability of success for each trial is 0.5, as if flipping a fair coin.

The binomial probability mass function (PMF), denoted as P(x=k), calculates the probability that the random variable takes exactly k successes in n trials. It encompasses the binary nature of the outcomes, the independence between trials, and the fixed number of trials, and hence, is inherently linked to combinatorial mathematics, which allows the calculation of possible combinations of outcomes.

Combinatorial Mathematics

Combinatorial mathematics plays a fundamental role in calculating probabilities within discrete probability distributions, especially for binomial random variables. It involves the study of countable, discrete structures and allows for the computation of how many different ways things can be arranged or combined.

In the context of binomial probabilities, we use the combinatorial function \( \binom{n}{k} \) known as 'n choose k', which represents the number of ways to select k successes out of n trials, without regard for order. This mathematical concept is critical in building the binomial PMF because it helps account for every possible combination that could result in k successes.

To calculate \( \binom{n}{k} \) we use the factorial function. The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. The combinatorial function is defined as \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). This relationship is essential in determining probabilities of specific outcomes for binomial random variables and is a powerful tool in many areas of probability and statistics.

Principle of Complementarity

The principle of complementarity is a simple yet profound concept utilized in probability theory. This principle states that the probability of an event not occurring is equal to one minus the probability of the event occurring. Mathematically, for any event A, the complementarity principle is expressed as:\[ P(\text{not } A) = 1 - P(A) \].

In the exercise at hand, to find the probability that a binomial random variable 'x' is greater than 1, we first calculate the probability of the complementary event, that is 'x' being less than or equal to 1. By summing these complementary probabilities and subtracting from one, we leverage the principle of complementarity to determine our desired probability. This method is particularly useful when it is easier to calculate the probability of the opposite or complementary event, often simplifying the problem-solving process.

Applying this principle to our binomial problem not only streamlines our calculations but also provides insight into the relationship between an event and its complement, illustrating the interconnectedness of probabilities within a given sample space.