In a certain population of wookies, the probability of having blue eyes is 0.30. 50% of all wookies have outward-poking belly buttons ("outies"). It also turns out that of all wookies with outies, 40% of them have blue eyes. What is the probability that a blue-eyed wookie will have an outie? Round your answer to the hundredths place (e.g., 0.XX).
Question
In a certain population of wookies, the probability of having blue eyes is 0.30. 50% of all wookies have outward-poking belly buttons ("outies"). It also turns out that of all wookies with outies, 40% of them have blue eyes. What is the probability that a blue-eyed wookie will have an outie? Round your answer to the hundredths place (e.g., 0.XX).
Solution
To solve this problem, we need to use Bayes' theorem, which is a way to find a probability when we know certain other probabilities. The formula is:
P(A|B) = P(B|A) * P(A) / P(B)
Where:
- P(A|B) is the probability we are trying to find: the probability that a wookie has an outie given that it has blue eyes.
- P(B|A) is the probability that a wookie has blue eyes given that it has an outie, which we know is 0.40.
- P(A) is the probability that a wookie has an outie, which we know is 0.50.
- P(B) is the probability that a wookie has blue eyes, which we know is 0.30.
Substituting these values into the formula, we get:
P(A|B) = 0.40 * 0.50 / 0.30
Doing the multiplication and division, we get:
P(A|B) = 0.20 / 0.30 = 0.67
So, the probability that a blue-eyed wookie has an outie is approximately 0.67, or 67%, when rounded to the hundredths place.
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