If [tex]P(A)=2 / 3[/tex], [tex]P(B)=3 / 5[/tex] and [tex]P(A \cap B)=2 / 5[/tex], find [tex]P(A \cup B)[/tex].
a. [tex]\frac{13}{15}[/tex]
b. [tex]\frac{12}{15}[/tex]
c. [tex]\frac{7}{15}[/tex]
d. None of them
Answer (2)
We calculated the probability of the union of events A and B using the formula P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) . Substituting the given probabilities, we found that P ( A ∪ B ) = 15 13 , which corresponds to option a.
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Use the formula for the probability of the union of two events: P ( A B ) = P ( A ) + P ( B ) − P ( A B ) .
Substitute the given values: P ( A ) = 3 2 , P ( B ) = 5 3 , and P ( A B ) = 5 2 .
Calculate the result: P ( A B ) = 3 2 + 5 3 − 5 2 = 15 10 + 15 9 − 15 6 = 15 13 .
The final answer is: 15 13 .
Explanation
Understand the problem We are given the probabilities of events A and B, as well as the probability of their intersection, and we are asked to find the probability of their union.
State the formula We will use the formula for the probability of the union of two events: P ( A B ) = P ( A ) + P ( B ) − P ( A B ) .
Substitute the values We are given that P ( A ) = 3 2 , P ( B ) = 5 3 , and P ( A B ) = 5 2 . Substituting these values into the formula, we get: P ( A B ) = 3 2 + 5 3 − 5 2 .
Find common denominator Now, we need to find a common denominator to add and subtract the fractions. The least common denominator of 3 and 5 is 15. So we rewrite the fractions with the common denominator: 3 2 = 35 25 = 15 10 5 3 = 53 33 = 15 9 5 2 = 53 23 = 15 6 .
Substitute equivalent fractions Now we can substitute these equivalent fractions into the equation: P ( A B ) = 15 10 + 15 9 − 15 6 .
Calculate the result Now we perform the addition and subtraction: 15 10 + 15 9 = 15 19 15 19 − 15 6 = 15 13 .Therefore, P ( A B ) = 15 13 .
Final Answer The probability of A B is 15 13 . Comparing this to the given options, we see that option a is 15 13 .
Examples
Understanding the probability of the union of events is crucial in many real-world scenarios. For example, consider a marketing campaign where event A is a customer clicking on an ad and event B is a customer making a purchase. Knowing P ( A ) , P ( B ) , and P ( A B ) allows marketers to calculate P ( A B ) , which represents the probability that a customer either clicks on the ad or makes a purchase (or both). This helps in evaluating the effectiveness of the campaign and optimizing strategies to increase customer engagement and sales. The formula P ( A B ) = P ( A ) + P ( B ) − P ( A B ) is fundamental in such analyses.
