To calculate the probability that your professor picks the 2 questions you study for out of the 3 questions they choose, we use the concept of combinations in probability.

Step 1: Total combinations

The total number of ways to choose 3 questions from the 5 available is given by:

$\binom{5}{3} = \frac{5!}{3!(5-3)!} = 10$

Step 2: Favorable combinations

For your professor to select the 2 questions you studied for, those 2 questions must be in the selection. This leaves 1 more question to be chosen from the remaining 3 questions (5 total - 2 studied). The number of favorable ways this can happen is:

$\binom{3}{1} = 3$

Step 3: Probability calculation

The probability of the professor selecting the 2 questions you studied for is the ratio of favorable combinations to total combinations:

$P = \frac{\text{Favorable combinations}}{\text{Total combinations}} = \frac{3}{10} = 0.3$

Final Answer:

The probability is 30%.

Would you like more details or a breakdown of these steps?

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Here are 5 related questions to consider:

1. What if your professor chooses 4 questions instead of 3?
2. How does the probability change if you study for 3 out of the 5 questions?
3. What is the probability that none of your studied questions are chosen?
4. Can the probability be increased by studying an additional question?
5. How many combinations exist if the professor has 10 questions and picks 4?

Tip: When solving probability problems, always start by identifying the total and favorable outcomes using combinations.