Answer:
To find the value of \( p \) (the probability of a success) in a binomial distribution, you typically follow these steps:
1. **Identify the Total Number of Trials (\( n \)):** Determine how many times the experiment is conducted or how many trials are performed.
2. **Determine the Number of Successes (\( k \)):** Count the number of successful outcomes you are interested in.
3. **Calculate the Probability of a Single Success (\( p \)):** This is the probability of a success in a single trial, which can often be determined from the context of the problem. If the problem does not provide this directly, you may need to use historical data or experimental results to estimate it.
4. **Calculate the Probability of a Single Failure (\( q \)):** This is \( 1 - p \), the probability of not having a success in a single trial.
For example, if you are flipping a fair coin 10 times and you want to find the probability of getting heads (a success), \( p \) is 0.5 (since the probability of getting heads on a fair coin flip is 0.5).
### Formula for Binomial Probability
To find the probability of getting exactly \( k \) successes in \( n \) trials, you can use the binomial probability formula:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
where:
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \)
- \( p \) is the probability of a single success
- \( (1-p) \) is the probability of a single failure
### Example
Suppose you conduct a survey where you ask 100 people if they like a new product, and historically, 70% of people like the product. You want to find the probability that exactly 75 out of 100 people will say they like the product. Here:
- \( n = 100 \)
- \( k = 75 \)
- \( p = 0.7 \)
- \( q = 1 - p = 0.3 \)
Using the binomial probability formula:
\[ P(X = 75) = \binom{100}{75} (0.7)^{75} (0.3)^{25} \]
This calculation would give you the probability of exactly 75 successes (people liking the product) out of 100 trials (survey responses).