A bag contains fifty slips (small pieces of paper) with the numbers 1, 2, . . . , 50 written on them

(One number on each slip!). Your friend writes down ten numbers from 1, 2, . . . , 50 on a sheet
of paper, and then you draw—without replacement—twenty slips randomly from the bag, and
look at the numbers drawn.
1. What is the sample space?
2. What is the probability that all the numbers which your friend wrote down also appear
among the drawn slips?
3. What is the probability that none of your friend’s numbers is drawn?
4. What is the probability that exactly half of your friend’s numbers are drawn?

To answer these questions, we first need to understand some basic probability concepts.

1. Sample Space:
The sample space is the set of all possible outcomes of an experiment. In this case, the experiment is drawing twenty slips from the bag, and the sample space is all the different combinations of twenty slips that can be drawn from fifty slips. The size of the sample space can be calculated using binomial coefficients: 50 choose 20, denoted as C(50,20), which equals 2118760.

2. Probability that all of your friend's numbers appear:
To calculate this probability, we need to consider the favorable outcomes, which are the combinations where all of your friend's numbers appear among the twenty slips drawn. The number of favorable outcomes can be calculated by choosing ten slips from the remaining forty slips, denoted as C(40,10), which equals 847,660. Therefore, the probability is the ratio of favorable outcomes to the sample space: P(all numbers are drawn) = 847660/2118760 ≈ 0.400

3. Probability that none of your friend's numbers are drawn:
To calculate this probability, we need to consider the unfavorable outcomes, which are the combinations where none of your friend's numbers appear among the twenty slips drawn. The number of unfavorable outcomes can be calculated by choosing twenty slips from the remaining forty slips, denoted as C(40,20), which equals 137,846,528. Therefore, the probability is the ratio of unfavorable outcomes to the sample space: P(no numbers are drawn) = 137846528/2118760 ≈ 0.065

4. Probability that exactly half of your friend's numbers are drawn:
To calculate this probability, we need to consider the favorable outcomes, which are the combinations where exactly ten slips out of your friend's ten numbers are among the twenty slips drawn. The number of favorable outcomes can be calculated by choosing ten slips from your friend's ten numbers and ten slips from the remaining forty slips, denoted as C(10,10) * C(40,10), which equals 102,722,781,440. Therefore, the probability is the ratio of favorable outcomes to the sample space: P(exactly half numbers are drawn) = 102722781440/2118760 ≈ 48.46

Remember, these calculations are based on the assumption that the slips are drawn randomly without replacement.

1. The sample space consists of all possible combinations of 20 slips drawn from the bag. Since there are 50 slips in the bag, the size of the sample space is given by the binomial coefficient "50 choose 20" denoted as C(50, 20).

2. To calculate the probability that all the numbers your friend wrote down appear among the drawn slips, we need to consider two things:
a) The number of successful outcomes - the number of ways to choose 20 slips such that all of your friend's numbers are included. This is given by the binomial coefficient "10 choose 10," denoted as C(10, 10).
b) The total number of possible outcomes - the total number of ways to choose 20 slips from the bag. This is given by the binomial coefficient "50 choose 20," denoted as C(50, 20).

The probability is then calculated as the number of successful outcomes divided by the total number of possible outcomes:

P(all friend's numbers appear) = C(10, 10) / C(50, 20)

3. To calculate the probability that none of your friend's numbers are drawn, we need to consider two things:
a) The number of successful outcomes - the number of ways to choose 20 slips such that none of your friend's numbers are included. This is given by the binomial coefficient "40 choose 20," denoted as C(40, 20).
b) The total number of possible outcomes - the total number of ways to choose 20 slips from the bag. This is given by the binomial coefficient "50 choose 20," denoted as C(50, 20).

The probability is then calculated as the number of successful outcomes divided by the total number of possible outcomes:

P(no friend's numbers appear) = C(40, 20) / C(50, 20)

4. To calculate the probability that exactly half of your friend's numbers are drawn, we need to consider two things:
a) The number of successful outcomes - the number of ways to choose 10 slips out of the 20 drawn slips such that exactly 5 of them match your friend's numbers. This is given by the binomial coefficient "10 choose 5," denoted as C(10, 5).
b) The total number of possible outcomes - the total number of ways to choose 20 slips from the bag. This is given by the binomial coefficient "50 choose 20," denoted as C(50, 20).

The probability is then calculated as the number of successful outcomes divided by the total number of possible outcomes:

P(half of friend's numbers appear) = C(10, 5) / C(50, 20)