Question

Consider a binomial random variable with \(n=8\) and \(p=.7 .\) Let \(x\) be the number of successes in the sample. Evaluate the probabilities in Exercises \(2-6 .\) $$ P(x=3) $$

Short answer

Answer: The probability of having exactly 3 successes in 8 trials with a success probability of 0.7 is approximately 0.02907.

Step-by-step solution

Identify given values

We are given the values \(n=8\), \(p=0.7\), and \(x=3\). Now, let's plug these values into the binomial probability formula.

Calculate the binomial coefficient

We need to calculate the binomial coefficient \(\binom{n}{k} = \binom{8}{3}\). We can do this using the formula: $$ \binom{8}{3} = \frac{8!}{3!(8-3)!}=\frac{8!}{3!5!} $$ Now, let's calculate the factorials of 3 and 5: $$ 3! = 3 * 2 * 1 = 6 $$ $$ 5! = 5 * 4 * 3 * 2 * 1 = 120 $$ So the binomial coefficient becomes: $$ \binom{8}{3} = \frac{8!}{6 * 120} $$ Now calculate the factorial of 8: $$ 8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40320 $$ Finally, let's compute the binomial coefficient: $$ \binom{8}{3} = \frac{40320}{6 * 120} = 56 $$

Compute the probability

Now, let's calculate the probability using the binomial probability formula: $$ P(x=3) = \binom{8}{3} \cdot 0.7^3 \cdot (1-0.7)^{8-3} $$ Plug in the binomial coefficient we calculated in Step 2: $$ P(x=3) = 56 \cdot 0.7^3 \cdot (0.3)^{5} $$ Now, let's calculate the probabilities: $$ 0.7^3 = 0.7 * 0.7 * 0.7 \approx 0.3429 $$ $$ 0.3^5 = 0.3 * 0.3 * 0.3 * 0.3 * 0.3 \approx 0.00243 $$ Now, let's plug these values into the probability equation: $$ P(x=3) = 56 * 0.3429 * 0.00243 \approx 0.02907 $$ So, the probability of having exactly 3 successes in the sample is approximately \(0.02907\).

Key concepts

Binomial Random Variable

A binomial random variable is a specific type of discrete random variable that counts how often a particular event occurs in a fixed number of trials, or attempts. In the context of binomial probability, each trial is independent of the others, and there are only two possible outcomes for each trial, often labeled as 'success' and 'failure'.

The probability of success, denoted as 'p', remains constant for each trial. For the given exercise, the number of trials (n) is 8, and the probability of success for each trial (p) is 0.7. When you're asked to find the probability that a binomial random variable 'x' equals a certain value, you're computing the chance that the event will occur that specific number of times out of the total trials.

Understanding binomial random variables assists in predicting outcomes for processes that match this structure, such as the likelihood of getting a certain number of heads in coin tosses, or the probability of a basketball player hitting a set number of free throws in a series of attempts.

Binomial Coefficient

The binomial coefficient, often symbolized as \binom{n}{k}\text{ or }\( C(n, k) \), is a fundamental part of calculating binomial probabilities. It represents the number of ways to choose 'k' successes out of 'n' trials, regardless of order, which is crucial in working out the binomial distribution formula.

To calculate the binomial coefficient, you use the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Factorials play a key role here, where \( n! \) (n factorial) is the product of all positive integers up to 'n'. As shown in the solution, calculating the binomial coefficient for \( \binom{8}{3} \) involves factorials of 8, 3, and 5. The simplicity of the formula allows for a clear calculation of possible combinations of successes and failures within a set of trials. The computation of the binomial coefficient is integral for determining exact probabilities of events under a binomial distribution.

Factorial Calculation

A factorial, denoted by an exclamation point (!), is the product of all positive integers less than or equal to a particular number. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials are deeply connected to the concept of permutations and combinations, serving as building blocks for various mathematical concepts such as binomial coefficients.

To enhance understanding, let's explore a straightforward example. If you're asked to find \( 3! \), you simply multiply 3 by every positive integer below it, so \( 3! = 3 \times 2 \times 1 = 6 \). The solution of the exercise requires factorial calculations when determining the binomial coefficient. It's worth noting when computing factorials that \( 0! = 1 \), a useful mathematical convention especially when dealing with binomial coefficients where \( k \) or \( n-k \) may be zero.

Factorial calculation is a fundamental skill in mathematics, especially in statistics and probability. It enables clear and precise computation of combination numbers which are crucial for solving binomial probability problems.