Answer:
Let's break down each part of this problem step by step:
a. How many different lab groups are possible?
To calculate the total number of different lab groups possible, we need to consider all possible combinations of lab groups with at least one engineer. You can use the principle of inclusion-exclusion here.
First, calculate the total number of ways to form lab groups without any restrictions (4 people chosen from 12):
Total Ways = C(12, 4)
Now, calculate the number of ways to form lab groups with no engineers:
Ways without Engineers = C(5, 4)
So, the total number of lab groups with at least one engineer is:
Total Lab Groups = Total Ways - Ways without Engineers
b. What is the probability that a lab group contains three engineers?
To find the probability of a lab group containing three engineers, we'll first calculate the total number of ways to choose three engineers from seven engineers and one non-engineer from five non-engineers. Then we'll divide this by the total number of lab groups we calculated in part (a).
Probability = (C(7, 3) * C(5, 1)) / Total Lab Groups
c. Given that you are an engineer and your friend is not an engineer, what is the probability that you will be assigned to a lab group with your friend?
In this case, there are two engineers (you and your friend) and two non-engineers left to choose from. You want to calculate the probability that both you and your friend are in the same lab group.
Probability = (1 way to choose you) * (1 way to choose your friend) / Total Lab Groups
d. Given that you and a second friend are both engineers, what is the probability that you will be assigned to a lab group with your second friend?
In this case, there are two engineers (you and your second friend) and three non-engineers left to choose from. You want to calculate the probability that both you and your second friend are in the same lab group.
Probability = (1 way to choose you) * (1 way to choose your second friend) / Total Lab Groups
Now, let's calculate these probabilities:
a. Total Lab Groups:
Total Ways = C(12, 4) = 495
Ways without Engineers = C(5, 4) = 5
Total Lab Groups = 495 - 5 = 490
b. Probability of containing three engineers:
Probability = (C(7, 3) * C(5, 1)) / Total Lab Groups
Probability = (35 * 5) / 490
Probability = 175 / 490 = 35/98
c. Probability of being in the same group as your non-engineer friend:
Probability = (1 * 1) / Total Lab Groups
Probability = 1 / 490
d. Probability of being in the same group as your second engineer friend:
Probability = (1 * 1) / Total Lab Groups
Probability = 1 / 490
So, the probabilities are:
b. 35/98
c. 1/490
d. 1/490
Explanation: