70% of new employees take a learning lessons.

During first month of work they have probability of 0.04 to make mistakes while those who didn't take lessons (30%) have probability of 0.09.
Given that an employee didn't make mistakes in his first month, what is the probability that he had token lessons

look at 1000 new employees

700 took lessons
... 28 made mistakes in 1st month

300 did not take lessons
... 27 made mistakes in 1st month

[1000 - (28 + 27)] did not make a mistake

[700 - 28] took lessons

200

To find the probability that an employee had taken lessons given that they didn't make any mistakes in their first month, we can use Bayes' theorem.

Let's denote the events as:
A: Employee took lessons
B: Employee didn't make mistakes in the first month

We are given:
P(A) = 0.70 (Probability of an employee taking lessons)
P(~A) = 0.30 (Probability of an employee not taking lessons)
P(B|A) = 0.04 (Probability of an employee making mistakes given they took lessons)
P(B|~A) = 0.09 (Probability of an employee making mistakes given they didn't take lessons)

We need to find P(A|B) (Probability of an employee taking lessons given they didn't make mistakes).

Using Bayes' theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)

We can calculate P(B) using the Law of Total Probability:
P(B) = P(B|A) * P(A) + P(B|~A) * P(~A)

Substituting the given values:
P(A|B) = (0.04 * 0.70) / [(0.04 * 0.70) + (0.09 * 0.30)]

Calculating this expression:
P(A|B) = 0.028 / (0.028 + 0.027)

Simplifying:
P(A|B) = 0.028 / 0.055
P(A|B) = 0.5091

Therefore, the probability that the employee had taken lessons given they didn't make any mistakes in their first month is approximately 0.5091, or 50.91%.