Question

Consider a binomial random variable with \(n=9\) and \(p=.3 .\) Let \(x\) be the number of successes in the sample. Evaluate the probabilities in Exercises \(7-10 .\) The probability that \(x\) is less than 2 .

Short answer

Answer: The probability that the number of successes is less than 2 is approximately 0.3945.

Step-by-step solution

Understand the binomial formula

The binomial formula is given as: \(P(x=k)=\binom{n}{k}p^{k}(1-p)^{n-k}\) Here, \(n\) is the number of trials, \(p\) is the probability of success in each trial, and \(k\) is the number of successes.

Calculate the probability of zero successes

We will calculate the probability of having zero successes (i.e. \(x=0\)): \(P(x=0)=\binom{n}{0}p^{0}(1-p)^{n-0}=\binom{9}{0}0.3^{0}(1-0.3)^{9-0}\) Using the properties of binomial coefficients, we get \(\binom{9}{0}=1\). So, the calculation becomes: \(P(x=0)=(1)(1)(0.7)^{9}=0.7^9\)

Calculate the probability of one success

We will calculate the probability of having one success (i.e. \(x=1\)): \(P(x=1)=\binom{n}{1}p^{1}(1-p)^{n-1}=\binom{9}{1}0.3^{1}(1-0.3)^{9-1}\) Using the properties of binomial coefficients, we get \(\binom{9}{1}=9\). So, the calculation becomes: \(P(x=1)=(9)(0.3)(0.7)^{8}\)

Calculate the probability of \(x\) being less than 2

Now we add the probabilities calculated in steps 2 and 3: \(P(x<2)=P(x=0)+P(x=1)=0.7^9+(9)(0.3)(0.7)^{8}\) After calculating the probabilities, we get: \(P(x<2) \approx 0.1211+0.2734 \approx 0.3945\) So, the probability that \(x\) is less than 2 is approximately \(0.3945\).

Key concepts

Binomial Probability Distribution

The binomial probability distribution is a fundamental concept in statistics associated with random variables representing the number of successes in a fixed number of independent trials, each with the same probability of success. When flipping a coin, rolling a dice, or examining quality control failures, you're often dealing with binomial situations if the trials are independent and the probability of success is constant.

In the context of our exercise, we're considering a binomial random variable where each trial is the performance of an action with two possible outcomes: success (with probability \( p \) ) or failure (with probability \( 1-p \) ). The variable \( x \) represents the total number of successes in \( n \) trials, which, for our example, are 9 in total with a success probability of 0.3.

Binomial Formula

Understanding the binomial formula is akin to unlocking a math treasure chest that holds the secrets to predicting outcomes. It is succinctly expressed as <\(P(x=k)=\binom{n}{k}p^k(1-p)^{n-k}\)>.In this treasure of a formula, \(\binom{n}{k}\) represents the number of ways to choose \(k\) successes out of \(n\) trials, also known as 'n choose k' or the binomial coefficient. This part of the formula is essential for calculating the different possible outcomes of \(k\) successes. The succeeding terms \(p^k\) and \( (1-p)^{n-k} \) account for the likelihood of those successes and failures occurring in the trials.
Let's turn this magical formula into reality with our exercise example, where \(n=9\) and \(p=0.3\). Applying our understanding, we calculate the probabilities for exactly 0 and 1 successes, translating the abstract into a concrete chance of occurrence.

Probability Calculation

Probability calculation is the engine room of binomial distribution. Once you comprehend the binomial formula, performing probability calculations is like following a recipe – you simply plug in the values and compute. For our exercise, the goal was to find the probability that \(x\) is less than 2. This means we need to sum the probabilities of \(x\) being 0 and 1, as calculated in the steps provided.

We follow a straightforward process: multiply, power, and sum. First, we find the probability of zero successes (\(0.7^9\)) and one success (\(9 * 0.3 * 0.7^8\)). Then, we add these probabilities together to find the total probability of \(x\) being less than 2. Mastery of these calculations allows for the accurate prediction of occurrences in binomial distributions, an invaluable skill for anyone delving into statistics and probability theory.