Conditional Probabilities and Independence of Events
The probability of event a occurring given that event b has occurred is 20%. Similarly, the probability of event b occurring given that event a has occurred is 25%. How to find? Are a and b independent events?
Find the conditional probabilities of events a and b. Are events a and b independent?
Conditional Probabilities and Independence of Events
To find the conditional probabilities, we use the formula: p(a|b) = p(a and b)/p(b) and p(b|a) = p(a and b)/p(a)
Substituting the given probabilities, we get:
p(a|b) = 3%/15% = 20%
p(b|a) = 3%/12% = 25%
To find the probability of a given b, we use the formula: p(a and b)/p(b) = 3%/15% = 20%
Since the probability of a given b is not the same as the probability of a, we can say that a and b are not independent events.
Detail Explanation of Conditional Probabilities and Independence of Events
Conditional probabilities are used to determine the likelihood of an event happening given that another event has already occurred. In this case, the probability of event a occurring given that event b has occurred is 20%, while the probability of event b occurring given that event a has occurred is 25%.
To calculate the conditional probabilities, we divide the probability of both events happening by the probability of the second event. Substituting the given probabilities, we find that the conditional probabilities of a given b and b given a are 20% and 25% respectively.
When the probability of a given b is not the same as the probability of a, we conclude that events a and b are not independent. This means that the occurrence of one event affects the likelihood of the other event happening.
Understanding conditional probabilities and independence of events is crucial in probability theory and helps in making informed decisions based on the relationship between different events.