Question

Evaluate the binomial probabilities in Exercises \(16-19\). $$ C_{1}^{7}(.2)^{1}(.8)^{6} $$

Short answer

Answer: The probability of obtaining exactly 1 success out of 7 trials in this binomial experiment is approximately 0.3667.

Step-by-step solution

Calculate binomial coefficient

First, we need to calculate the number of ways to choose 1 success out of 7 trials using the binomial coefficient formula which is presented as follows: $$ C_{k}^{n} = \frac{n!}{k!(n-k)!} $$ In our exercise, \(n=7\) and \(k=1\), so the binomial coefficient will be calculated as: $$ C_{1}^{7} = \frac{7!}{1!(7-1)!} $$

Evaluate the binomial coefficient

Now we'll compute the factorials: $$ C_{1}^{7} = \frac{7!}{1!(6!)} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(1)(6 \times 5 \times 4 \times 3 \times 2 \times 1)} = 7 $$ Hence, there are 7 ways to get exactly 1 success in 7 trials.

Calculate the binomial probability

Now, we'll use the binomial probability formula: $$ P(X=1) = C_{1}^{7}(.2)^{1}(.8)^{6} $$ Plugging in the values we found earlier, as well as the given values for \(p\) and \(1-p\): $$ P(X=1) = 7(.2)^{1}(.8)^{6} $$

Evaluate the binomial probability

Finally, we'll calculate the probability as follows: $$ P(X=1) = 7(.2)(.8)^{6} = 7 \times 0.2 \times (\approx 0.2621) = \approx 0.3667 $$ Thus, the probability of obtaining exactly 1 success out of 7 trials in this binomial experiment is approximately 0.3667.

Key concepts

Binomial Coefficient

The concept of a binomial coefficient is central to understanding binomial probability. It is represented by the notation \(C_k^n\) and is a quantitative expression of the number of ways to choose \(k\) successes in \(n\) trials. It forms the combinatorial basis for calculating the likelihood of a specific number of successes in a sequence of independent trials, each with two possible outcomes.

To calculate the binomial coefficient, you use the following formula: \[C_k^n = \frac{n!}{k!(n-k)!}\]. Here, the exclamation point denotes a factorial, indicating the product of all positive integers up to that number. In our exercise, we computed \(C_1^7\) to find there are 7 ways to achieve exactly one success in seven trials. Understanding this concept is crucial because it differentiates between different outcomes where order does not matter - a fundamental aspect in combinatorics and probability.

Factorial Computation

Factorials are integral in both permutations and combinations and thus, in computing binomial coefficients. A factorial, denoted by an exclamation point \(n!\), is the product of all positive integers less than or equal to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

In the step-by-step solution, factorial computation simplified the binomial coefficient calculation. We evaluated \(7!\) and divided it by \(1! \(7 - 1\)!\) which simplifies to \(7\). Understanding how to efficiently calculate factorials, often by canceling common terms in the numerator and denominator, reduces the complexity of finding binomial coefficients, and thus, improves problem-solving efficiency.

Probability Theory

Probability theory is the mathematical framework for quantifying uncertainty. It provides methods to calculate the likelihood of events within a well-defined set of possibilities. In the context of our exercise, we applied probability theory to a binomial experiment.

The formula for binomial probability is given by \[ P(X=k) = C_k^n(p)^k(1-p)^{(n-k)} \], where \(P(X=k)\) is the probability of \(k\) successes, \(C_k^n\) is the binomial coefficient, \(p\) is the probability of success on a single trial, and \(1-p\) is the probability of failure.
Utilizing the relevant values, we calculated the probability for exactly one success out of seven trials. This application of probability theory is vital for predicting outcomes, and it extends to various disciplines including finance, science, engineering, and more. It's important to grasp these probabilities to make informed decisions based on expected outcomes.