In a certain population, the probability that a woman has red hair is 1/4. If a woman has red hair, the probability of her having freckles is 3/4. If she does not have red hair, the probability of her having freckles is 1/16 .
Find the following probabilities. Enter your final answers in percent notation rounded to the nearest tenth.
(a) If a woman is chosen at random from the population, what is the probability that she will have freckles? Hint: women may have freckles regardless of the color of their hair.
Answer?
%
(b) If a woman will have freckles, what is the probability that she has red hair?
Answer?
%
Make a tree-diagram, two main brances as R and NR , (red and non-red)
each of those has sub-brances of F and NF
place the probabilities on each branch
you have 4 possible outcomes
R-F = (1/4)(3/4) = 3/16
R-NR = (1/4)(1/4) = 1/16
NR-F = (3/4)(1/16) = 3/64
NR-NF = (3/4)(15/16) = 45/64
And there you have it! , just convert to percentages,
notice the 4 outcomes add up to 1
To find the probabilities, we can use the formula for conditional probability:
P(A|B) = P(A and B) / P(B)
where A and B are events, P(A|B) is the probability of event A occurring given that event B has occurred, P(A and B) is the probability of both events A and B occurring, and P(B) is the probability of event B occurring.
(a) To find the probability that a woman chosen at random will have freckles, we can use the law of total probability. We need to consider two cases:
1) Woman has red hair (event A)
2) Woman does not have red hair (event A')
Using the given probabilities:
P(freckles) = P(freckles | red hair) * P(red hair) + P(freckles | not red hair) * P(not red hair)
= (3/4) * (1/4) + (1/16) * (3/4)
= 3/16 + 3/64
= 19/64 ≈ 29.7%
Therefore, the probability that a woman chosen at random will have freckles is approximately 29.7%.
(b) To find the probability that a woman has red hair given that she has freckles, we can use Bayes' theorem:
P(red hair | freckles) = P(freckles | red hair) * P(red hair) / P(freckles)
Using the given probabilities:
P(red hair | freckles) = (3/4) * (1/4) / (19/64)
= 3/16 * 64/19
= 12/19 ≈ 63.2%
Therefore, the probability that a woman has red hair given that she has freckles is approximately 63.2%.
To find the probabilities, we will use the multiplication rule and the law of total probability.
(a) To find the probability that a woman chosen at random will have freckles, we can consider two cases: either she has red hair or she does not have red hair.
Case 1: She has red hair.
The probability of choosing a woman with red hair is 1/4, and the probability of a woman with red hair having freckles is 3/4. So, the probability that a woman with red hair has freckles is (1/4) * (3/4) = 3/16.
Case 2: She does not have red hair.
The probability of choosing a woman without red hair is 3/4, and the probability of a woman without red hair having freckles is 1/16. So, the probability that a woman without red hair has freckles is (3/4) * (1/16) = 3/64.
Now, using the law of total probability, we add up the probabilities from the two cases:
Probability of having freckles = (3/16) + (3/64) = 15/64.
To express this probability as a percentage, we divide 15 by 64 and multiply by 100:
Probability of having freckles = (15/64) * 100 ≈ 23.4%.
Therefore, the probability that a woman chosen at random from the population will have freckles is approximately 23.4%.
(b) To find the probability that a woman with freckles has red hair, we only need to consider the case where the woman has freckles.
Given that a woman has freckles, there are two possibilities: either she has red hair or she does not have red hair.
Case 1: She has red hair.
The probability of a woman having red hair and freckles is (1/4) * (3/4) = 3/16.
Case 2: She does not have red hair.
The probability of a woman not having red hair and still having freckles is (3/4) * (1/16) = 3/64.
Using the law of total probability, we add up the probabilities from the two cases:
Probability of having freckles with red hair = 3/16.
Probability of having freckles without red hair = 3/64.
Now, to find the probability that a woman with freckles has red hair, we divide the probability of having freckles with red hair by the total probability of having freckles:
Probability of having freckles with red hair / Probability of having freckles = (3/16) / (15/64).
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
Probability of having freckles with red hair / Probability of having freckles = (3/16) * (64/15) ≈ 256/80.
Simplifying the fraction, we get:
Probability of having freckles with red hair ≈ 3.2.
To express this probability as a percentage, we multiply by 100:
Probability of having freckles with red hair ≈ 3.2 * 100 ≈ 320%.
Therefore, the probability that a woman with freckles has red hair is approximately 320%.