3 statistics professors and 7 chemistry professors are available to be advisors.If each professor has an equal chance of being selected, what is the probability that both professors are cgemistry professors?
To find the probability that both professors are chemistry professors, we need to determine the total number of possible outcomes and the number of favorable outcomes.
First, let's calculate the total number of possible outcomes. We have a total of 10 professors available (3 statistics professors + 7 chemistry professors). Therefore, there are 10 professors to choose from.
Next, let's calculate the number of favorable outcomes. We want to choose 2 chemistry professors from the available 7. This can be calculated using the concept of combinations. We can use the formula for combinations:
C(n, k) = n! / (k! * (n-k)!)
where n represents the total number of objects to choose from and k represents the number of objects to be chosen.
In this case, we want to find C(7, 2), which can be calculated as follows:
C(7, 2) = 7! / (2! * (7-2)!)
= 7! / (2! * 5!)
= (7 * 6 * 5!) / (2! * 5!)
= 7 * 6 / 2!
= 21
Therefore, there are 21 favorable outcomes where both professors are chemistry professors.
Now, let's calculate the probability:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 21 / 10
= 2.1
Hence, the probability that both professors selected are chemistry professors is 2.1. However, probabilities cannot exceed 1, so the correct probability would be 1 (or 100%) in this case.
To find the probability that both professors are chemistry professors, we need to calculate the probability of selecting a chemistry professor for the first advisor and then selecting another chemistry professor for the second advisor.
There are 7 chemistry professors out of a total of 10 professors (3 statistics + 7 chemistry). So the probability of selecting a chemistry professor for the first advisor is 7/10.
After selecting a chemistry professor for the first advisor, there are 6 chemistry professors remaining out of a total of 9 professors (2 statistics + 6 chemistry). So the probability of selecting a chemistry professor for the second advisor, given that the first advisor is a chemistry professor, is 6/9.
To find the probability of both events occurring (selecting a chemistry professor for the first and second advisors), we multiply the probabilities:
(7/10) * (6/9) = 42/90 = 7/15
Therefore, the probability that both professors are chemistry professors is 7/15.