Question

1. the probability that a randomly chosen male has a blood circulation problem is 0.25. males who have a blood circulation problem are twice as likely to be smokers as those who do not have a blood circulation problem.

calculate the probability that a male has a blood circulation problem, given that he is a smoker.

(a) 1/4
(b) 1/3
(c) 2/5
(d) 1/2
(e) 2/3

Explanation:

⚡ Using what you learned: Conditional Probability · 🆕 New: Bayes' Theorem
Reversing conditional probabilities using known branch rates

Step 1: Define events and given probabilities

Let \( C \) be the event that a male has a blood circulation problem, and \( S \) be the event that a male is a smoker.

From the problem statement:

$$ P(C) = 0.25 $$

$$ P(C') = 1 - 0.25 = 0.75 $$

We are given that males with a circulation problem are twice as likely to be smokers as those without:

$$ P(S \mid C) = 2 \cdot P(S \mid C') $$

Let \( P(S \mid C') = p \). Then:

$$ P(S \mid C) = 2p $$

Step 2: Apply Bayes' Theorem

We want to find the conditional probability that a male has a circulation problem given that he is a smoker, \( P(C \mid S) \):

$$ P(C \mid S) = \frac{P(S \mid C) P(C)}{P(S)} $$

First, express the total probability of being a smoker, \( P(S) \):

$$ P(S) = P(S \mid C) P(C) + P(S \mid C') P(C') $$

$$ P(S) = (2p)(0.25) + (p)(0.75) $$

$$ P(S) = 0.5p + 0.75p = 1.25p $$

Step 3: Calculate the final probability

Substitute the values back into the Bayes' Theorem formula:

$$ P(C \mid S) = \frac{(2p)(0.25)}{1.25p} $$

$$ P(C \mid S) = \frac{0.5p}{1.25p} = \frac{0.5}{1.25} = \frac{2}{5} $$

Answer:

(C) 2/5