Question

Evaluate the binomial probabilities in Exercises \(16-19\). $$ C_{2}^{8}(.3)^{2}(.7)^{6} $$

Short answer

Answer: The value of the binomial probability expression is approximately \(0.294237\).

Step-by-step solution

Calculate the number of combinations

In this case, we have \(n = 8\) and \(x = 2\). Using the formula for calculating the number of combinations: $$ C_{8}^{2} = \frac{8!}{2!(8-2)!} = \frac{8!}{2!6!} $$ Compute the factorials: $$ 8! = 8\times7\times6\times5\times4\times3\times2\times1 = 40320 $$ $$ 2! = 2\times1 = 2 $$ $$ 6! = 6\times5\times4\times3\times2\times1 = 720 $$ Now, compute the combination: $$ C_{8}^{2} = \frac{40320}{2\times720} = 28 $$

Evaluate the given expression

Now that we have the value of \(C_{8}^{2}\), we can replace it in the given expression: $$ 28(.3)^{2}(.7)^{6} $$ Evaluate the powers: $$ (.3)^{2} = 0.09,\quad(.7)^{6} = 0.117649 $$ Substitute these values back into the expression: $$ 28\times0.09\times0.117649 $$ And finally, evaluate the expression: $$ 28\times0.09\times0.117649 \approx 0.294237 $$ So, the value of the given binomial probability is approximately \(0.294237\).

Key concepts

Combinations and Permutations

When dealing with probability and statistics, understanding the difference between combinations and permutations is critical. Combinations refer to the selection of items where the order does not matter, whereas permutations are all about the order of selection.

For example, if you are choosing two fruits from a basket of apples, bananas, and cherries, the combination would be the pair regardless of whether you picked an apple first or a banana. In a permutation, picking an apple then a banana is different from picking a banana then an apple.

To calculate combinations, we use the formula \(C_n^x = \frac{n!}{x!(n - x)!}\), where 'n' is the total number of items, 'x' is the number of items being chosen, and '!' denotes a factorial. The factorial function, symbolized by an exclamation mark, indicates the product of all positive integers up to a given number.

Factorials in Probability

Factorials play a key role in probability, especially when evaluating the number of different ways things can be arranged or combined. A factorial, represented by the symbol \(n!\), is the product of all positive integers less than or equal to \(n\). For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

In probability, factorials are used to determine the number of possible outcomes. For example, in our exercise, to find out how many different ways we can select 2 successes from 8 trials, we use the factorial function as part of the combination formula to calculate \(C_8^2\).

The calculation of large factorials can be simplified using a calculator or computer software, as they can grow very large quite quickly. Most importantly, the concept of 0! is defined to be 1, which can be useful in calculating combinations and permutations where all items are chosen or none are.

Evaluating Expressions in Statistics

Evaluating expressions is a fundamental skill in statistics, where it often involves handling factorials, powers, and coefficients. In the context of the binomial probability, as in our exercise, the expression represents the probability of getting a certain number of successes \(x\) in a given number of trials \(n\), with a success probability \(p\) and failure probability \(q = 1 - p\).

To accurately evaluate these expressions, you need to perform several steps. First, calculate the necessary combinations to determine the number of ways to achieve \(x\) successes. Then, raise the probabilities to the respective powers of successes and failures. Last, multiply the combination by the calculated probabilities.

A sound understanding of this process is essential for correctly interpreting statistical data and conducting various probability-related calculations. Remember that precision is crucial, as small errors in calculation can lead to incorrect conclusions.