In a large crowd,there are three times as many men as women.
Three people are chosen at random. Assuming there are so many people that choosing three has a negligible effect on the proportion of men to women. Using tree probability.
Find the probability that they are
A. All men
B.2 women and 1 man
P (man)=1/4
P (man,man,man)=1/4)^3
P (2w,1m)
To find the probability of different outcomes, we need to calculate the total number of possible outcomes and the number of favorable outcomes for each scenario. Let's go step-by-step:
Given information:
- In a large crowd, there are three times as many men as women.
A. Probability of selecting all men:
To calculate this probability, we need to find the ratio of the number of favorable outcomes (selecting all men) to the total number of outcomes.
1. Find the ratio of men to women:
Given that there are three times as many men as women, let's assume there are x women in the crowd. Therefore, the number of men in the crowd would be 3x.
2. Find the total number of people in the crowd:
The total number of people in the crowd would be the sum of men and women, which is 3x + x = 4x.
3. Calculate the probability of selecting all men:
To select all men, we need to choose 3 men from the total number of men.
The total number of men in the crowd is 3x. We can use the formula for combinations: nCr = n! / (r!(n-r)!), where n is the total number of items, and r is the number of items chosen.
The probability of selecting all men is given by: (3x)C3 / (4x)C3.
B. Probability of selecting 2 women and 1 man:
To calculate this probability, we need to find the ratio of the number of favorable outcomes (selecting 2 women and 1 man) to the total number of outcomes.
1. Calculate the probability of selecting 2 women:
To select 2 women, we need to choose 2 women from the total number of women, which is x.
The probability of selecting 2 women is given by: xC2 / (4x)C3.
2. Calculate the probability of selecting 1 man:
To select 1 man, we need to choose 1 man from the total number of men, which is 3x.
The probability of selecting 1 man is given by: (3x)C1 / (4x)C3.
3. Calculate the probability of selecting 2 women and 1 man:
To find the probability of selecting 2 women and 1 man, we multiply the probabilities of selecting 2 women and 1 man together:
Probability = (xC2 / (4x)C3) * ((3x)C1 / (4x)C3).
Note: Since the problem specifies that choosing three people has a negligible effect on the proportion of men to women, the values of x will cancel out when calculating the probabilities.
By simplifying the expressions further, you should be able to obtain the answers in terms of x, and if necessary, substitute the value for x to get numerical probabilities.
To find the probability in this scenario, we can use tree diagrams. Let's proceed step by step:
Step 1: Determine the total number of people in the crowd:
Let's assume there are x women in the crowd. Since there are three times as many men as women, the number of men in the crowd would be 3x. Therefore, the total number of people in the crowd would be x + 3x = 4x.
Step 2: Calculate the probability of choosing all men:
Since there are 3x men and 4x people in total, the probability of choosing a man on the first pick would be 3x/4x = 3/4. After selecting a man, there would be 3x - 1 men and 4x - 1 people remaining. The probability of selecting a man on the second pick, considering one man has already been chosen, would be (3x - 1)/(4x - 1). Similarly, the probability of selecting a man on the third pick would be (3x - 2)/(4x - 2). To find the probability of choosing all men, we multiply these three probabilities together:
Probability of choosing all men = (3/4) * ((3x - 1)/(4x - 1)) * ((3x - 2)/(4x - 2))
Step 3: Calculate the probability of choosing 2 women and 1 man:
Similarly, we can calculate the probability of choosing 2 women and 1 man. The probability of selecting a woman on the first pick would be x/4x = 1/4. After selecting a woman, there would now be x - 1 women and 4x - 1 people remaining. The probability of selecting a woman on the second pick, considering one woman has already been chosen, would be (x - 1)/(4x - 1). The probability of selecting a man on the third pick would be (3x)/(4x - 2). To find the probability of choosing 2 women and 1 man, we multiply these three probabilities together:
Probability of choosing 2 women and 1 man = (1/4) * ((x - 1)/(4x - 1)) * (3x/(4x - 2))
Now, if we are given the value of x (the number of women in the crowd), we can substitute it into these formulas to find the numerical probabilities. However, without knowing the value of x, we cannot find the exact probabilities.