Question

A population consists of 25 men and 25 women. A simple random sample (draws at random without replacement) of 4 people is chosen. Find the chance that in the sample:
1-there are more women than men
2-the third person is a woman, given that the first person and fourth person are both men

Answer 1

If I understood correctly, without replacement means that after choosing someone, you cannot choose them again, so I will solve according to this.
Let's do this in steps.
1. We pick the first person and take them out of the room. They can be of any gender.
2. We pick the second person. They must be of the same gender of the first person. Since the first person is no longer in the room, there are now 49 people in the room, with 24 people of the first person's gender, such that the probability that the second person is of the same gender as the first is 2449
3. We pick the third person, out of the room of now 48 people. There are now only 23 people of the same gender as the first person in the room, such that the probability that the third person is of the same gender as the first and second is 2348.
4. Now, we pick the fourth person of a room of 47 people, which contains 22 people of the same gender as the first person, such that the probability that the fourth person will be of the same gender as the first person is 2247.
Now, we want all of these events (meaning 1,2,3,4) to occur, and therefore, we need to multiply these probabilities, in order to get that the probability that all of the people are of the same gender is 24⋅23⋅2249⋅48⋅47.
In general, using this algorithm, we can find a formula for the probability to choose k people of the same gender from a room of n men and n women.
The probability that the second person's gender is the same as the first's is 24/49 (there are 24 unchosen people of that gender left and 49 unchosen people in total). The probability that the third person's gender is the same again is 23/48, and the probability for the fourth is 22/47.
(24×23×22)/(49×48×47)=2532303≈0.11.

Answer 2

If I understood correctly, without replacement means that after choosing someone, you cannot choose them again, so I will solve according to this.
Let's do this in steps.
1. We pick the first person and take them out of the room. They can be of any gender.
2. We pick the second person. They must be of the same gender of the first person. Since the first person is no longer in the room, there are now 49 people in the room, with 24 people of the first person's gender, such that the probability that the second person is of the same gender as the first is 24/49
3. We pick the third person, out of the room of now 48 people. There are now only 23 people of the same gender as the first person in the room, such that the probability that the third person is of the same gender as the first and second is 23/48.
4. Now, we pick the fourth person of a room of 47 people, which contains 22 people of the same gender as the first person, such that the probability that the fourth person will be of the same gender as the first person is 22/47.
Now, we want all of these events (meaning 1,2,3,4) to occur, and therefore, we need to multiply these probabilities, in order to get that the probability that all of the people are of the same gender is 24⋅23⋅22/49⋅48⋅47.
In general, using this algorithm, we can find a formula for the probability to choose k people of the same gender from a room of n men and n women.
The probability that the second person's gender is the same as the first's is 24/49 (there are 24 un-chosen people of that gender left and 49 un-chosen people in total). The probability that the third person's gender is the same again is 23/48, and the probability for the fourth is 22/47.
(24×23×22)/(49×48×47)=253/2303≈0.11.