Question

Evaluate the binomial probabilities in Exercises \(16-19\). $$ C_{0}^{4}(.05)^{0}(.95)^{4} $$

Short answer

Answer: The binomial probability is approximately 0.8145.

Step-by-step solution

Calculating the Combination

To calculate the combination \(C_{0}^{4}\), we'll use the formula: $$ C_{n}^{r} = \frac{n!}{r!(n-r)!} $$ In our case, \(n = 4\) and \(r = 0\). So, we have: $$ C_{0}^{4} = \frac{4!}{0!(4-0)!} $$ Since \(0! = 1\) and \(4! = 4 \times 3 \times 2 \times 1 = 24\), we get: $$ C_{0}^{4} = \frac{24}{1 \times 24} = 1 $$

Calculating the Probability of Each Event

In the binomial probability expression, we have: $$ (.05)^{0}(.95)^{4} $$ We need to raise \(0.05\) to the power of \(0\) and \(0.95\) to the power of \(4\). So, we have: $$ (.05)^{0} = 1 $$ $$ (.95)^{4} = (.95) \times (.95) \times (.95) \times (.95) \approx 0.8145 $$

Finding the Binomial Probability

Now we need to multiply the number of combinations by the probability of each event: $$ C_{0}^{4}(.05)^{0}(.95)^{4} = 1 \times 1 \times 0.8145 \approx 0.8145 $$ So, the binomial probability is approximately \(0.8145\).

Key concepts

Combinations in Probability

When tackling probability problems, it's crucial to understand how to count outcomes efficiently. Combinations are used when the order of the outcomes does not matter, and is represented by the notation . The general formula for the number of combinations of r elements from a set of n elements is .
This represents the number of ways to pick r unordered outcomes from n possibilities. An essential part of calculating combinations is understanding factorial calculation, which is the product of all positive integers up to a certain number. For example, means multiplying every whole number from 4 down to 1.
Improving your understanding can be done by practicing various combinations with different values of n and r. It helps to remember that , as this means that there is only one way to choose nothing from a set, and similarly, choosing all elements from the set in a particular order has only one combination.

Binomial Distribution

In probability and statistics, a very common scenario is the binomial distribution, which arises when you repeat an experiment with two outcomes (a 'success' and a 'failure') for a fixed number of times. The key feature of a binomial distribution is that each trial is independent of the others. The probability of success is represented by p and the probability of failure by q (where q = 1 - p).

Calculating Binomial Probabilities

The probability of getting exactly r successes in n trials is given by the formula . Here, the term calculates the different ways in which the successes can be distributed among the trials, while and denote the probabilities of having all successes and all failures, respectively. It's vital to understand binomial distribution for analyzing scenarios where only two possible outcomes exist, such as tossing a coin, checking for defective items in a batch, or determining the likelihood of events in a sports game.
To better understand binomial distribution, consider working on exercises with varying probabilities p and different number of trials n, and see how these changes impact the distribution.

Factorial Calculation

The factorial is a fundamental concept when working with combinations, permutations, and other areas of mathematics such as calculus and series expansion. The factorial of a non-negative integer n, denoted by , is the product of all positive integers less than or equal to n. For instance, .
A unique and important property of factorial is . This fact becomes especially useful in simplifying expressions where factorials appear in both the numerator and denominator, which is often the case in combination and probability calculations. Factorials grow at a very fast rate, so even with small numbers, the values can become quite large.
When improving upon the basics of factorial calculation, it can be helpful to visualize factorials as the number of ways to arrange a set of objects. Also, practicing with factorial calculations will not only sharpen your probability skills but will also help you to understand concepts in other areas that use combinatorial arguments or series expansions.