Conditional Probability and Bayes' Theorem

Bayes' Theorem is a fundamental formula in probability theory that allows you to calculate the conditional probability of an event A given that another event B has already occurred.

Using Bayes' Theorem, we can calculate the probability of a person having the disease given that the test result is negative as follows: P(Disease|Negative Test) = P(Negative Test|Disease) * P(Disease) / P(Negative Test) = 0.1 * 0.05 / (1 - 0.9) = 0.005.

To calculate the probability, we need to consider the proportion of women and men in the town and their respective probabilities of owning a car. The probability of a randomly selected person owning a car can be calculated as follows: P(Owns Car) = P(Woman and Owns Car) + P(Man and Owns Car) = (0.6 * 0.7) + (0.4 * 0.3) = 0.42 + 0.12 = 0.52.

To calculate the probability of traffic congestion, we need to consider the probability of rain and the respective probabilities of traffic congestion given rain or no rain. The probability of traffic congestion can be calculated as follows: P(Traffic Congestion) = P(Rain and Traffic Congestion) + P(No Rain and Traffic Congestion) = (0.3 * 0.7) + (0.7 * 0.2) = 0.21 + 0.14 = 0.34.

To calculate the probability that a defective product came from Line A, we need to consider the proportion of output from each line and their respective defect rates. The probability can be calculated as follows: P(Defective from Line A) = P(Line A and Defective) / P(Defective) = (0.6 * 0.02) / (0.6 * 0.02 + 0.4 * 0.04) = 0.012 / 0.016 = 0.72.

To calculate the probability that a person prefers only coffee, we need to subtract the proportion of people who prefer both coffee and tea from the proportion of people who prefer coffee. The probability can be calculated as follows: P(Prefers Only Coffee) = P(Prefers Coffee) - P(Prefers Both) = 0.7 - 0.1 = 0.6.

To calculate the probability that a person who tests positive actually has the disease, we need to consider the sensitivity and specificity of the test, as well as the prevalence of the disease. The probability can be calculated using Bayes' Theorem as follows: P(Disease|Positive Test) = P(Positive Test|Disease) * P(Disease) / P(Positive Test) = 0.99 * 0.01 / (0.99 * 0.01 + 0.05 * 0.99) = 0.94.

To calculate the probability that a person owns a car or a bike, we need to consider the proportion of people who own a car, a bike, or both. The probability can be calculated as follows: P(Owns Car or Bike) = P(Owns Car) + P(Owns Bike) - P(Owns Both) = 0.6 + 0.3 - 0.1 = 0.8.

To calculate the probability that a defective product came from Line B, we need to consider the proportion of output from each line and their respective defect rates. The probability can be calculated as follows: P(Defective from Line B) = P(Line B and Defective) / P(Defective) = (0.3 * 0.02) / (0.7 * 0.05 + 0.3 * 0.02) = 0.006 / 0.008 = 0.2.

To calculate the probability that a person prefers only tea, we need to subtract the proportion of people who prefer both coffee and tea from the proportion of people who prefer tea. The probability can be calculated as follows: P(Prefers Only Tea) = P(Prefers Tea) - P(Prefers Both) = 0.3 - 0.2 = 0.1.

To calculate the probability that a person who tests positive actually has the disease, we need to consider the sensitivity and specificity of the test, as well as the prevalence of the disease. The probability can be calculated using Bayes' Theorem as follows: P(Disease|Positive Test) = P(Positive Test|Disease) * P(Disease) / P(Positive Test) = 0.98 * 0.02 / (0.98 * 0.02 + 0.1 * 0.98) = 0.86.