8.0235% of the employees of the Acme Tire store are enrolled in a 401k at the store. If the company has 600 employees, what is the probability that less than 40 employees are entered in the 401k.
To find the probability that less than 40 employees are enrolled in the 401k, we need to use the binomial distribution. The binomial distribution is used when there are two possible outcomes (success or failure) for a fixed number of independent trials. In this case, the success is an employee being enrolled in the 401k, and the failure is an employee not being enrolled.
The formula for the binomial distribution probability is:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes,
- C(n, k) is the number of combinations of n items taken k at a time,
- p is the probability of success on a single trial, and
- (1-p) is the probability of failure on a single trial.
In our case, n = 600 (total number of employees), p = 0.080235 (probability of an employee being enrolled in the 401k), and we want to find the probability of less than 40 successes (employees enrolled).
Now, let's calculate the probability:
P(X < 40) = P(X = 0) + P(X = 1) + ... + P(X = 39)
To calculate each term, we can use the binomial distribution formula. However, calculating each term separately can be time-consuming. So, instead, we can use the cumulative distribution function (CDF) of the binomial distribution to calculate the probability directly.
The CDF calculates the probability of getting up to a certain number of successes. In our case, we want the probability of getting up to 39 successes.
P(X < 40) = CDF(39) = Σ P(X = k) for k from 0 to 39
To calculate this probability using a calculator or statistical software, you need to find the cumulative probability of the binomial distribution with the given parameters (n = 600, p = 0.080235) up to 39 successes.