A binomial setting involves repeated trials of a random process, where the following conditions are met

• two possible outcomes (success/failure)
• independent trials of same random process
• a fixed number of trials , $n$
• each trial has the same probability of success, $p$

In a binomial setting, the random variable X, which equals the number of successes, is called the binomial random variable.

The probability distribution of X is called the binomial distribution.

<aside> ⭐ The probability of getting exactly x successes in n trials is

$P(X=x) = \binom nx p^x(1-p)^{n-x}$

You can also just use binompdf(n, p, x)

</aside>

Example

The probability a tropical storm becomes a hurricane is 0.53. If the weather service predicts 6 more tropical storms, what is the probability that exactly 5 of them become hurricanes?

• Two outcomes
  • "success" = hurricane, "failure" = not hurricane
• Independent trials
  • Knowing one tropical storm's outcome does not help us predict another tropical storm's outcome.
• A fixed number of trials
  • n = 6 tropical storms
• Each trial has the same probability of success
  • p = 0.53

This satisfies all the conditions for the binomial distribution.