Let's analyze the problem step by step.

Given Data:

• Total number of students: 120
• Haskayne School of Business students (HS): 54
• 1st-Year students (1st): 71
• 1st-Year students who are not in the Haskayne School: 20

From the above, we can conclude the number of 1st-Year students in the Haskayne School: $\text{HS} \cap \text{1st} = 71 - 20 = 51$

We can now use this information to complete the table, find the required probabilities, and answer the questions.

Part (a) - Complete the Table:

We need to fill in the joint and marginal probabilities. First, let's calculate the number of students in each category:

1. HS and 1st (HS ∩ 1st): 51
2. HS and not 1st (HS ∩ 1stᶜ): 54 - 51 = 3
3. Not HS and 1st (HSᶜ ∩ 1st): 20
4. Not HS and not 1st (HSᶜ ∩ 1stᶜ): 120 - (51 + 3 + 20) = 46

Using these, we can calculate the probabilities by dividing by the total number of students, 120:

Part (b) - Probability the student is either Haskayne or 1st-Year:

The event "Haskayne or 1st-Year" is the union of the two events: $P(HS \cup 1st) = P(HS) + P(1st) - P(HS \cap 1st)$ $P(HS \cup 1st) = 0.4500 + 0.5917 - 0.4250 = 0.6167$

Part (c) - Probability the student is neither Haskayne nor 1st-Year:

This event corresponds to the complement of "Haskayne or 1st-Year": $P(\text{neither HS nor 1st}) = 1 - P(HS \cup 1st) = 1 - 0.6167 = 0.3833$

Part (d) - Proportion of 1st-Year students who are not in the Haskayne School:

This is simply the probability of being in the intersection of HSᶜ and 1st: $P(HSᶜ \cap 1st) = 0.1667$

Part (e) - Are $HS$ and $1st$ mutually exclusive?

Mutually exclusive events have no overlap, i.e., $P(HS \cap 1st) = 0$. However, in this case: $P(HS \cap 1st) = 0.4250 \neq 0$ Thus, HS and 1st are not mutually exclusive. The correct option is B.

Part (f) - Are $HS$ and $1st$ independent?

Two events are independent if: $P(HS \cap 1st) = P(HS) \times P(1st)$ Let's check: $P(HS) \times P(1st) = 0.4500 \times 0.5917 = 0.2663$ Since: $P(HS \cap 1st) = 0.4250 \neq 0.2663$ Thus, HS and 1st are not independent. The correct option is D.

Would you like further details or clarifications on any of these solutions?

Here are 5 related questions to deepen your understanding:

1. How would the table change if the total number of students increased to 150?
2. What is the probability of randomly selecting a student who is either in 1st-Year or not in the Haskayne School?
3. If a student is in the 1st year, what is the probability that they are also in the Haskayne School?
4. How would independence between $HS$ and $1st$ be determined using conditional probabilities?
5. What proportion of Haskayne students are in their 1st year?

Tip: When events are independent, the occurrence of one does not influence the probability of the other.