Questions: A recent census found that 51.9% of adults are female, 10.3% are divorced, and 6.3% are divorced females. For an adult selected at random, let F be the event that the person is female, and D be the event that the person is divorced. a. Obtain P(F), P(D), and P(F D). b. Determine P(F or D), and interpret your answer in terms of percentages. c. Find the probability that a randomly selected adult is male. P( male )= (Type an integer or a decimal. Do not round.)

A recent census found that 51.9% of adults are female, 10.3% are divorced, and 6.3% are divorced females. For an adult selected at random, let F be the event that the person is female, and D be the event that the person is divorced. a. Obtain P(F), P(D), and P(F D). b. Determine P(F or D), and interpret your answer in terms of percentages. c. Find the probability that a randomly selected adult is male. P( male )= (Type an integer or a decimal. Do not round.)
Transcript text: A recent census found that $51.9 \%$ of adults are female, $10.3 \%$ are divorced, and $6.3 \%$ are divorced females. For an adult selected at random, let $F$ be the event that the person is female, and $D$ be the event that the person is divorced. a. Obtain $P(F), P(D)$, and $P(F \& D)$. b. Determine $P(F$ or $D)$, and interpret your answer in terms of percentages. c. Find the probability that a randomly selected adult is male. $P($ male $)=$ $\square$ (Type an integer or a decimal. Do not round.)
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Solution

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Solution Steps

To solve the given problem, we need to interpret the probabilities from the given percentages and use basic probability rules to find the required probabilities.

a. The probabilities \( P(F) \), \( P(D) \), and \( P(F \& D) \) are directly given as percentages in the problem statement. Convert these percentages to probabilities by dividing by 100.

b. To find \( P(F \text{ or } D) \), use the formula for the probability of the union of two events: \[ P(F \text{ or } D) = P(F) + P(D) - P(F \& D) \]

c. The probability that a randomly selected adult is male is the complement of the probability that the adult is female. Use the formula: \[ P(\text{male}) = 1 - P(F) \]

Step 1: Calculate \( P(F) \), \( P(D) \), and \( P(F \cap D) \)

From the given data:

  • \( P(F) = 0.519 \)
  • \( P(D) = 0.103 \)
  • \( P(F \cap D) = 0.063 \)
Step 2: Calculate \( P(F \text{ or } D) \)

Using the formula for the union of two events: \[ P(F \text{ or } D) = P(F) + P(D) - P(F \cap D) \] Substituting the values: \[ P(F \text{ or } D) = 0.519 + 0.103 - 0.063 = 0.559 \]

Step 3: Calculate \( P(\text{male}) \)

The probability that a randomly selected adult is male is given by: \[ P(\text{male}) = 1 - P(F) \] Substituting the value: \[ P(\text{male}) = 1 - 0.519 = 0.481 \]

Final Answer

  • \( P(F) = 0.519 \)
  • \( P(D) = 0.103 \)
  • \( P(F \cap D) = 0.063 \)
  • \( P(F \text{ or } D) = 0.559 \)
  • \( P(\text{male}) = 0.481 \)

Thus, the final answers are: \[ \boxed{P(F) = 0.519} \] \[ \boxed{P(D) = 0.103} \] \[ \boxed{P(F \cap D) = 0.063} \] \[ \boxed{P(F \text{ or } D) = 0.559} \] \[ \boxed{P(\text{male}) = 0.481} \]

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