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Answer :
To solve this problem, we'll use Bayes' Theorem. Let's break down the steps:
1. Identify the Given Probabilities:
- The probability that a randomly selected woman is taller than 68 inches is 0.25.
- The probability that a randomly selected man is taller than 68 inches is 0.85.
- The probability that a randomly selected person from the population is a woman is 0.55.
- The probability that a randomly selected person from the population is a man is 0.45.
2. Calculate the Total Probability that a Person is Taller than 68 inches:
To do this, we will consider both men and women:
- Probability that a woman is taller than 68 inches is 0.25, and women make up 55% of the population.
- Probability that a man is taller than 68 inches is 0.85, and men make up 45% of the population.
We calculate the total probability of selecting someone taller than 68 inches:
[tex]\[
P(\text{Taller}) = (0.25 \times 0.55) + (0.85 \times 0.45) = 0.1375 + 0.3825 = 0.52
\][/tex]
3. Use Bayes' Theorem to Find the Probability that the Individual is a Woman Given They are Taller than 68 inches:
Bayes' Theorem formula:
[tex]\[
P(\text{Woman | Taller}) = \frac{P(\text{Taller | Woman}) \times P(\text{Woman})}{P(\text{Taller})}
\][/tex]
Plug in the values:
[tex]\[
P(\text{Woman | Taller}) = \frac{0.25 \times 0.55}{0.52} = \frac{0.1375}{0.52} \approx 0.2644
\][/tex]
So, the probability that a randomly selected individual who is taller than 68 inches is a female is approximately 0.2644, or 26.44%.
1. Identify the Given Probabilities:
- The probability that a randomly selected woman is taller than 68 inches is 0.25.
- The probability that a randomly selected man is taller than 68 inches is 0.85.
- The probability that a randomly selected person from the population is a woman is 0.55.
- The probability that a randomly selected person from the population is a man is 0.45.
2. Calculate the Total Probability that a Person is Taller than 68 inches:
To do this, we will consider both men and women:
- Probability that a woman is taller than 68 inches is 0.25, and women make up 55% of the population.
- Probability that a man is taller than 68 inches is 0.85, and men make up 45% of the population.
We calculate the total probability of selecting someone taller than 68 inches:
[tex]\[
P(\text{Taller}) = (0.25 \times 0.55) + (0.85 \times 0.45) = 0.1375 + 0.3825 = 0.52
\][/tex]
3. Use Bayes' Theorem to Find the Probability that the Individual is a Woman Given They are Taller than 68 inches:
Bayes' Theorem formula:
[tex]\[
P(\text{Woman | Taller}) = \frac{P(\text{Taller | Woman}) \times P(\text{Woman})}{P(\text{Taller})}
\][/tex]
Plug in the values:
[tex]\[
P(\text{Woman | Taller}) = \frac{0.25 \times 0.55}{0.52} = \frac{0.1375}{0.52} \approx 0.2644
\][/tex]
So, the probability that a randomly selected individual who is taller than 68 inches is a female is approximately 0.2644, or 26.44%.
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