Final answer:
12% of employees are both researchers and have a data science diploma, while 40% of data science diploma holders are researchers. Additionally, 38% of employees are either researchers or have a data science diploma (or both), and the two events are not independent.
Step-by-step explanation:
To calculate the proportion of employees who are researchers and have a data science diploma, we use the given percentages. If 20% of the employees are researchers and 60% of these researchers have a data science diploma, we calculate the intersection of these two groups by multiplying the two percentages:
0.20 (researchers) × 0.60 (researchers who have a diploma) = 0.12 or 12%.
So, 12% of the employees are both researchers and have a data science diploma.
To find the proportion of data science diploma holders who are researchers, we need to calculate the conditional probability. Given that 30% of employees have a data science diploma, and we know that 12% of the total employees are both researchers and have a diploma, we calculate:
0.12 (both) ÷ 0.30 (diploma holders) = 0.40 or 40%.
So, 40% of the data science diploma holders are researchers.
To find the proportion of employees who are either researchers or have a data science diploma (or both), we use the formula:
P(A or B) = P(A) + P(B) - P(A and B), where A is the event of being a researcher, and B is the event of having a diploma:
0.20 (researchers) + 0.30 (diploma holders) - 0.12 (both) = 0.38 or 38%.
So, 38% of the employees are either researchers or have a data science diploma (or both).
To determine if the events of being a researcher and having a data science diploma are independent, we need to check if P(A and B) = P(A) × P(B). Here, P(A and B) is 0.12, but P(A) × P(B) is 0.20 × 0.30 = 0.06. Since 0.12 does not equal 0.06, the two events are not independent.