$A$ and $B$ are independent events.
[tex]
\begin{array}{l}
P(A)=0.50 \\
P(B)=0.20
\end{array}
[/tex]
What is $P(A \mid B)$ ?
A. 0.50
B. 0.10
C. Not enough information
D. 0.20
Answer (1)
Independent events A and B have probabilities P ( A ) = 0.50 and P ( B ) = 0.20 .
Calculate the joint probability: P ( A ∩ B ) = P ( A ) × P ( B ) = 0.50 × 0.20 = 0.10 .
Apply the conditional probability formula: P ( A ∣ B ) = P ( B ) P ( A ∩ B ) .
Find the result: P ( A ∣ B ) = 0.20 0.10 = 0.50 .
Explanation
Understand the problem and provided data We are given that events A and B are independent, with P ( A ) = 0.50 and P ( B ) = 0.20 . We want to find P ( A ∥ B ) , the conditional probability of A given B .
Use the definition of independent events Since A and B are independent events, the probability of both A and B occurring is the product of their individual probabilities: P ( A ∩ B ) = P ( A ) × P ( B ) .
Calculate the probability of A and B Substituting the given values, we have P ( A ∩ B ) = 0.50 × 0.20 = 0.10 .
Use the definition of conditional probability The conditional probability of A given B is defined as P ( A ∣ B ) = P ( B ) P ( A ∩ B ) .
Calculate the conditional probability Substituting the values we have, P ( A ∣ B ) = 0.20 0.10 = 0.50 .
State the final answer Therefore, P ( A ∣ B ) = 0.50 .
Examples
Understanding conditional probabilities is useful in many real-world scenarios. For example, in medical testing, it helps determine the probability of a patient having a disease given a positive test result, considering the test's accuracy and the prevalence of the disease in the population. In marketing, it can help predict the likelihood of a customer making a purchase given they have clicked on an advertisement. These calculations help in making informed decisions based on available data.
