Sure, let’s go through each variable in detail to clearly understand what they represent in the expression for calculating binomial probabilities:
1. [tex]\( n \)[/tex] represents the number of trials:
- In a binomial experiment, we perform a fixed number of independent trials.
- For example, if we are flipping a coin 10 times, [tex]\( n \)[/tex] would be 10.
2. [tex]\( p \)[/tex] represents the probability of success on a single trial:
- This is the likelihood that a single trial will result in success.
- For instance, if we are looking at the probability of getting a head in a single coin flip, and the coin is fair, [tex]\( p \)[/tex] would be 0.5.
3. [tex]\( k \)[/tex] represents the number of successes:
- This is the number of times we achieve the desired outcome (success) in our [tex]\( n \)[/tex] trials.
- Continuing with the coin flip example, if we want to know the probability of getting exactly 3 heads out of 10 flips, [tex]\( k \)[/tex] would be 3.
So the completed expression looks like this:
[tex]\[ n \text{ represents the } \text{number of trials} \][/tex]
[tex]\[ p \text{ represents the } \text{probability of success on a single trial} \][/tex]
[tex]\[ k \text{ represents the } \text{number of successes} \][/tex]
I hope this clarifies what each variable stands for in the binomial probability formula!
