Every afternoon, Ron enjoys a cup of hot tea in one of his many teacups. Once, Ron counted all the teacups in his collection, noting their colors and size.The probability that a cup is white is 0.9, the probability that it is small is 0.6, and the probability that it is white or small is 1.What is the probability that a randomly chosen cup is white and small?Write your answer as a whole number, decimal, or simplified fraction.
Question
Every afternoon, Ron enjoys a cup of hot tea in one of his many teacups. Once, Ron counted all the teacups in his collection, noting their colors and size.The probability that a cup is white is 0.9, the probability that it is small is 0.6, and the probability that it is white or small is 1.What is the probability that a randomly chosen cup is white and small?Write your answer as a whole number, decimal, or simplified fraction.
Solution
To find the probability that a randomly chosen cup is both white and small, we can use the formula for the probability of the union of two events:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
where:
- P(A ∪ B) is the probability that either event A occurs, or event B occurs, or they both occur,
- P(A) is the probability that event A occurs,
- P(B) is the probability that event B occurs,
- P(A ∩ B) is the probability that both events A and B occur.
In this case, event A is the cup being white, and event B is the cup being small. We are given that P(A) = 0.9, P(B) = 0.6, and P(A ∪ B) = 1. We want to find P(A ∩ B), the probability that a cup is both white and small.
Substituting the given values into the formula, we get:
1 = 0.9 + 0.6 - P(A ∩ B)
Solving for P(A ∩ B), we find:
P(A ∩ B) = 0.9 + 0.6 - 1 = 0.5
So, the probability that a randomly chosen cup is both white and small is 0.5, or 50%.
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