To calculate P(D|C), which is the probability of event D occurring given that event C has occurred, we will use Bayes' theorem. Bayes' theorem relates the conditional probabilities of two events.
Bayes' theorem is given by:
P(D|C) = P(C|D) * P(D) / P(C)
where:
- P(D|C) is the probability that event D occurs given that event C has occurred.
- P(C|D) is the probability that event C occurs given that event D has occurred.
- P(D) is the probability that event D occurs.
- P(C) is the probability that event C occurs.
Given the values:
- P(C) = 0.4
- P(D) = 0.5
- P(C|D) = 0.4
Let's calculate P(D|C) step by step:
P(D|C) = P(C|D) * P(D) / P(C)
P(D|C) = 0.4 * 0.5 / 0.4
We can simplify the equation by canceling out the common factor in the numerator and the denominator:
P(D|C) = 0.5
So, the probability of D given C, P(D|C), is 0.5. If you'd like your answer as a decimal rounded to the tenths place, it would be 0.5 since there is no need for any other digits after the decimal for this particular result.