Suppose that the population of a certain townis made up of 45% of men and 55% of women. Of the men, 40% wear glasses, abd of the women, 20% wear glasses. Given that a person chosen at random from the town wears glasses, what is the probability that the person is a woman?
Probability: 11 out of 29;
or 37.9310345%.
Method: Take a sample of 100.
45 are men; 40% or 18 men wear glasses
55 are women; 20% or 11 women wear glasses.
So out of 100, 18+11=29 wear glasses out of which 11 are women.
Probability: 11 out of 29
Well, let's crunch some numbers and have a little statistical fun, shall we?
First, let's assume we have a friendly town with 100 inhabitants. We have 45 men and 55 women, so that's 0.45 x 100 = 45 men and 0.55 x 100 = 55 women.
Out of these 45 jolly men, 40% wear glasses. That's 0.40 x 45 = 18 gentlemen wearing glasses.
Out of our fabulous 55 women, 20% wear glasses, resulting in 0.20 x 55 = 11 stylish ladies rocking glasses.
Oh no! I forgot to mention that the person chosen randomly wears glasses. So we don't have to worry about the people without glasses anymore. We are only concerned with the people who do have glasses.
Now, if you pick a citizen randomly who wears glasses, there are a total of 18 + 11 = 29 people to consider.
And since we only want to find the probability that the person chosen is a woman, we'll focus on the 11 magnificent women with glasses. So the probability you're looking for is 11 (women with glasses) divided by 29 (total number of people with glasses).
Calculating this delightful probability, we get 11/29 ≈ 0.38.
So, rejoice! The probability that a person chosen randomly from the town, who wears glasses, is a woman is approximately 0.38, or in simpler terms, around 38%.
Hope that brings a smile to your face! If not, I'll keep clowning around until it does! 🤡
To solve this problem, we will use Bayes' theorem.
Let's define the following events:
A: A person chosen at random wears glasses.
B: A person chosen at random is a woman.
We are given the following probabilities:
P(A|B') = 40% (40% of men wear glasses, so 60% of men don't wear glasses)
P(A|B) = 20% (20% of women wear glasses)
P(B) = 55% (55% of the population is made up of women)
Now, let's calculate the probability that a person chosen at random wears glasses, irrespective of their gender (P(A)):
P(A) = P(A|B)*P(B) + P(A|B')*P(B')
Substituting the given probabilities:
P(A) = 0.20*0.55 + 0.40*0.45
= 0.11 + 0.18
= 0.29
Finally, we can use Bayes' theorem to calculate the probability that the person wearing glasses is a woman (P(B|A)):
P(B|A) = (P(A|B)*P(B)) / P(A)
Substituting the given probabilities:
P(B|A) = (0.20*0.55) / 0.29
= 0.11 / 0.29
≈ 0.3793
Therefore, the probability that a person chosen at random from the town wears glasses and is a woman is approximately 0.3793, or 37.93%.
To find the probability that a person chosen at random from the town wears glasses and is a woman, we can use conditional probability.
Let's follow these steps to find the answer:
1. Determine the probabilities of each event:
- P(M) = probability of selecting a man = 45%
- P(W) = probability of selecting a woman = 55%
- P(G|M) = probability of a man wearing glasses = 40%
- P(G|W) = probability of a woman wearing glasses = 20%
2. Use the formula for conditional probability:
P(W|G) = (P(G|W) * P(W)) / P(G)
3. Plug in the values:
P(W|G) = (0.20 * 0.55) / P(G)
4. Find the probability of a person wearing glasses:
P(G) = (P(G|W) * P(W)) + (P(G|M) * P(M))
= (0.20 * 0.55) + (0.40 * 0.45)
Substitute the value of P(G) into the conditional probability formula:
P(W|G) = (0.20 * 0.55) / [(0.20 * 0.55) + (0.40 * 0.45)]
By calculating the expression, you can determine the probability that the person chosen at random from the town wears glasses and is a woman.