In a certain college , 4% of men students and 1% of women students are taller than 1.8m . Further more , 60% of the students are women . If a student is selected at random and is found taller than 1.8m , What is the probability that the student is a women?
To solve this problem, we can use Bayes' theorem:
Let's define the events:
A: The chosen student is a woman.
B: The chosen student is taller than 1.8m.
Given information:
P(A) = 0.60 (60% of the students are women)
P(A') = 1 - P(A) = 1 - 0.60 = 0.40 (the complement of event A)
P(B | A') = 0.04 (4% of men students are taller than 1.8m)
P(B | A) = 0.01 (1% of women students are taller than 1.8m)
We need to find P(A | B), which represents the probability that the student is a woman given that they are taller than 1.8m.
Using Bayes' theorem:
P(A | B) = (P(B | A) * P(A)) / P(B)
We can calculate P(B) using the law of total probability:
P(B) = P(B | A) * P(A) + P(B | A') * P(A')
Calculating P(B):
P(B) = (0.01 * 0.60) + (0.04 * 0.40)
= 0.006 + 0.016
= 0.022
Now we can substitute the known values into Bayes' theorem to find P(A | B):
P(A | B) = (0.01 * 0.60) / 0.022
= 0.006 / 0.022
= 0.273
Therefore, the probability that the student is a woman, given that they are taller than 1.8m, is approximately 0.273 or 27.3%.
To find the probability that a randomly selected student who is taller than 1.8m is a woman, we can use Bayes' theorem.
Bayes' theorem states that the probability of an event A given event B is equal to the probability of event B given event A, multiplied by the probability of event A, divided by the probability of event B.
Let's break down the information given in the question:
- 4% of men students are taller than 1.8m.
- 1% of women students are taller than 1.8m.
- 60% of the students are women.
Let's assign the following variables:
- M: event that a student is a man.
- W: event that a student is a woman.
- T: event that a student is taller than 1.8m.
We need to find the probability P(W|T), which represents the probability that a student is a woman given that they are taller than 1.8m.
Using Bayes' theorem, we have:
P(W|T) = P(T|W) * P(W) / P(T)
Now, let's calculate the probabilities:
- P(T|W) = 1% = 0.01 (as stated in the question)
- P(W) = 60% = 0.6 (as stated in the question)
- P(T) = P(T|W) * P(W) + P(T|M) * P(M)
- P(T|W) * P(W): 0.01 * 0.6 = 0.006
- P(T|M) * P(M): 0.04 * (1 - 0.6) = 0.016
- P(T) = 0.006 + 0.016 = 0.022
Now we can substitute the values into the formula:
P(W|T) = (0.01 * 0.6) / 0.022
Simplifying, we get:
P(W|T) ≈ 0.2727
Therefore, the probability that a randomly selected student who is taller than 1.8m is a woman is approximately 0.2727 or 27.27%.
Among 1000 random students (say), there will be 600 women of whom 6 (1%) are 1.8 meters or more high. There will be 400 men of whom 16 (4%) are that high.
Since the selection was made at random from the student population and ended up with one of the taller people, the chances it is a woman are
6/(16 + 6) = 27% That is the fraction of "tall people" that are women.
There is a way to do this using Bayes' theorem of conditional probabilities, which I can never remember, but it should give the same result.