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The probability formula for a binomial distribution is an essential tool for understanding how likely a specific outcome is. In a binomial distribution, we focus on the probability of a certain number of successes happening in a series of independent trials. The trials must be identical and must satisfy two conditions: each trial can only result in a success or a failure, and the probability of success remains constant across trials.

The formula used to calculate the probability of getting exactly \( k \) successes in \( n \) trials is:

• \( P(X=k) = \binom{n}{k} p^k q^{n-k} \)

Here:

• \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \)
• \( p \) is the probability of success on one trial
• \( q \) is the probability of failure on one trial, calculated as \( q = 1 - p \)

This formula helps in evaluating exact probabilities for various cases within a binomial framework.

In any problem involving a binomial distribution, it is crucial to correctly identify the parameters of the binomial distribution. These parameters help to define the structure of the problem and set the stage for calculations.

• The number of trials, denoted by \( n \), is the total number of times the experiment is conducted. In some questions, this could refer to the number of people surveyed, the number of times an event happens, etc. In our case, \( n = 16 \), since we are looking at 16 randomly selected individuals.
• The probability of success, denoted by \( p \), is the chance of one trial resulting in a success. It's crucial to know what success means in your context. Here, success means having high blood pressure, with a probability \( p = 0.20 \).
• The probability of failure, denoted by \( q \), is the chance of one trial failing. This is calculated as \( q = 1 - p \), which means \( q = 0.80 \) in this scenario.

These parameters are foundational and are always needed to utilize the probability formula successfully.

The binomial coefficient is a crucial component of the binomial probability formula. It indicates the number of ways to choose \( k \) successes from \( n \) trials without considering the order of successes.

The binomial coefficient is represented as \( \binom{n}{k} \) and can be computed using the formula:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Where \( n! \) (n factorial) is the product of all positive integers up to \( n \).

Example:

For instance, if we want to calculate \( \binom{16}{8} \), it tells us the number of ways to select 8 people out of 16 to have high blood pressure. Using the factorial method:

• \( \binom{16}{8} = \frac{16!}{8! \, 8!} = 12870 \). This value shows how many combinations are possible for this configuration.

Understanding and calculating the binomial coefficient correctly is critical as it directly impacts the probability results.

Calculating probabilities in a binomial distribution involves putting it all together: using the formula, parameters, and coefficients. You determine probabilities for various scenarios by substituting into the binomial probability formula.

Let's walk through two examples:

• Example 1: None have high blood pressure
  Substitute \( k = 0 \). Using the formula: \[ P(X=0) = \binom{16}{0} (0.20)^0 (0.80)^{16} \] \( \binom{16}{0} = 1 \), so the probability becomes \( 1 \times 1 \times 0.80^{16} \), approximately 0.028.
• Example 2: Exactly 4 have high blood pressure
  Substitute \( k = 4 \). Using the formula: \[ P(X=4) = \binom{16}{4} (0.20)^4 (0.80)^{12} \] \( \binom{16}{4} = 1820 \), so calculate \( 1820 \times (0.20)^4 \times (0.80)^{12} \) to get about 0.218.

By employing the formula consistently with the correct parameters, you can effectively determine and understand the likelihood of different outcomes occurring in a practical and reliable way.