Step-by-step solution:

I. Probability of a randomly selected student being accepted:

Let us denote:

• $p_1$: Probability of a student taking all 3 sciences and being accepted.
• $p_2$: Probability of a student taking 2 sciences and being accepted.
• $p_3$: Probability of a student taking 1 science and being accepted.
• $p_4$: Probability of a student taking no science and being accepted.

We will calculate this probability using the provided success rates for each group. Assume $x_1, x_2, x_3, x_4$ represent the probabilities of a randomly selected student taking 3 sciences, 2 sciences, 1 science, or no science, respectively. We multiply these by their success rates.

The formula is: $P(\text{accepted}) = (x_1 \times 0.85) + (x_2 \times 0.75) + (x_3 \times 0.20) + (x_4 \times 0.03)$

We need the distribution of students taking each number of sciences ($x_1, x_2, x_3, x_4$) to complete this calculation. If this data is not provided, let me know if I should make assumptions or proceed with additional information.

II. Likelihood of being accepted:

To determine if a randomly selected student is "likely" to be accepted, we compare $P(\text{accepted})$ with a threshold (e.g., 50%). If $P(\text{accepted}) \geq 0.5$, then it is likely. Otherwise, it is unlikely.

III. Recommendations to increase eligibility for at least 50 students:

To increase the likelihood that at least 50 students are eligible:

1. Increase science participation: Encourage more students to take 2 or 3 sciences, as these groups have higher success rates.

   • Evidence: Raising $x_1$ and $x_2$ increases their weighted contributions to the total probability.
   • Example: If the number of students in these groups is raised by 10%, the likelihood of acceptance improves significantly.

2. Improve success rates: Provide additional academic support to enhance the success rates in each group, especially in the 1- and 2-science categories.

   • Evidence: For instance, increasing the success rate for 2 sciences from 75% to 80% yields an improvement in the overall acceptance rate.

IV. Making sense of the context:

The answers will be evaluated based on:

1. The total distribution of students in the groups.
2. The realistic feasibility of increasing participation or success rates.

Would you like me to proceed with specific calculations based on assumed probabilities or clarify the data further? Let me know!

Expanding Questions:

1. How would altering the probabilities $x_1, x_2, x_3, x_4$ affect the overall acceptance rate?
2. What are potential strategies for increasing the success rates in each category?
3. How does the variability in group size impact the likelihood of a student being accepted?
4. Can a change in teaching methods improve the success rate for the single-science group?
5. How could external support (like tutoring programs) impact the overall acceptance probability?

Tip:

Always start probability questions by identifying the probabilities and weights of different groups to ensure accuracy when combining them.