In a large population, about  
45
%
  of people prefer tea over coffee. A researcher takes a random sample of  
13
  people and surveys whether they prefer tea over coffee. 

Use the binomial distribution to compute the probability that exactly  
6
  of the people in the sample prefer tea over coffee.

Identify the following information required to find the probability of people who prefer tea over coffee.

Provide your answer below:

n = 
 trials

x = 
 successes

p = 
 probability of those who prefer tea (as a decimal, not percent)

To solve this problem using the binomial distribution, we need to identify the following values:

- **n (number of trials)**: This is the total number of people surveyed.  
  \( n = 13 \)
  
- **x (number of successes)**: This is the number of people who prefer tea over coffee (i.e., the number of successes).  
  \( x = 6 \)
  
- **p (probability of success)**: This is the probability that a person prefers tea over coffee, given as a decimal. Since 45% prefer tea, we convert this to a decimal:  
  \( p = 0.45 \)

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Would you like me to compute the probability that exactly 6 people in the sample prefer tea over coffee?

### Here are 5 related questions:
1. What is the probability that fewer than 6 people prefer tea over coffee?
2. What is the probability that at least 6 people prefer tea over coffee?
3. How does increasing the sample size affect the probability?
4. What is the expected number of people who prefer tea in this sample?
5. What is the variance in the number of people who prefer tea?

### Tip:
In binomial distributions, the probability of success (p) and the number of trials (n) are key factors in determining outcomes. Adjusting either can significantly affect your results.

In a large population, about 45% of people prefer tea over coffee. A researcher takes a random sample of 13 people and surveys whether they prefer tea over coffee. Use the binomial distribution to compute the probability that exactly 6 of the people in the sample prefer tea over coffee.

Find out how to calculate the probability that exactly 6 out of 13 people prefer tea over coffee using the binomial distribution. This problem explores key probability concepts and uses the binomial probability formula. Perfect for high school students (Grades 9-12).

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