Question

suppose that 69% of the women who gave birth at a certain hospital last year were over 30 years old, and that 57% were unmarried. if 71% of the women were over 30 or unmarried (or both), what is the probability that a woman who gave birth at the hospital was both unmarried and over 30?

write your answer as a decimal (not as a percentage). (if necessary, consult a list of formulas.)

Explanation:

Define the events and given probabilities

Using the Probability of Events knowledge point, we define:

• Let \(A\) be the event that a woman who gave birth was over 30 years old.
• Let \(B\) be the event that a woman who gave birth was unmarried.

From the problem statement, we have:

• \(P(A) = 69\% = 0.69\)
• \(P(B) = 57\% = 0.57\)
• \(P(A \cup B) = 71\% = 0.71\) (representing over 30, unmarried, or both)

Apply the addition rule of probability

Using the Addition Rule of Probability knowledge point, the probability of the union of two events is:

$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$

Solve for the intersection probability

We rearrange the formula to solve for the probability that a woman was both unmarried and over 30, which is \(P(A \cap B)\):

$$ P(A \cap B) = P(A) + P(B) - P(A \cup B) $$

Substituting the given values:

$$ P(A \cap B) = 0.69 + 0.57 - 0.71 $$

$$ P(A \cap B) = 1.26 - 0.71 = 0.55 $$

Answer:

Suppose that 69% of the women who gave birth at a certain hospital last year were over 30 years old, and that 57% were unmarried. If 71% of the women were over 30 or unmarried (or both), what is the probability that a woman who gave birth at the hospital was both unmarried and over 30?

Write your answer as a decimal (not as a percentage).

<blank>0.55</blank>