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In this example, \( p(X = 3) \) is calculated because it answers a specific probability question within the context of a **binomial distribution** problem.

Here’s why \( P(X = 3) \) is considered:

1. **Problem Setup**: Six cola drinkers are given identical glasses containing cola \( S \) and cola \( F \). Each individual has an equal chance of preferring cola \( S \) or cola \( F \) because there’s no preference tendency. This makes the probability of choosing \( S \), \( p = 0.5 \).

2. **Binomial Distribution**: Since each choice is independent, and we are counting the number of people (out of six) who prefer \( S \), this situation fits a **binomial distribution**. The probability of success (choosing \( S \)) remains \( 0.5 \) for each trial.

3. **Choosing a Specific Outcome**: In this problem, they specifically ask about the probability of **exactly 3** out of the 6 drinkers preferring \( S \). This is denoted as \( X = 3 \), where \( X \) is the binomial random variable representing the number of drinkers who prefer \( S \).

4. **Calculation Focus**: The probability \( P(X = 3) \) is calculated rather than other values of \( X \) (like \( X = 4 \) or \( X = 5 \)) because the example focuses on the scenario of having exactly three drinkers who prefer cola \( S \). 

In summary, \( P(X = 3) \) is computed as the probability that exactly three of the six individuals choose cola \( S \), which is a direct application of the binomial probability formula for \( X = 3 \).

Would you like a step-by-step breakdown of how this probability \( P(X = 3) = 0.313 \) was calculated?

### Follow-up Questions:
1. What are the conditions for using a binomial distribution?
2. How would \( P(X = k) \) change if there was a preference tendency toward one cola?
3. What is the probability that all six drinkers prefer cola \( S \) (i.e., \( P(X = 6) \))?
4. How would you compute \( P(X = 2) \), the probability that exactly two people prefer cola \( S \)?
5. Why is the probability of each drinker choosing \( S \) exactly 0.5?

### Tip:
In binomial probability problems, clearly identifying \( n \), \( p \), and the exact outcome you’re interested in (like \( X = 3 \)) is key to applying the formula correctly.

Why is P(X = X) considered to be P(X = 3)?

Explore why P(X = 3) is calculated in a binomial distribution problem involving six cola drinkers, with equal probability of choosing between two colas. This example illustrates the application of the binomial probability formula for an exact outcome.

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