Question

What is the probability that at least one of them is selected?

Answer 1

Step 1: Likelihood of zero selections
The probability that no individuals are chosen is:\[P({No one selected}) = \left( 1 - \frac{1}{5} \right) \cdot \left( 1 - \frac{1}{3} \right) \cdot \left( 1 - \frac{1}{4} \right) = \frac{4}{5} \cdot \frac{2}{3} \cdot \frac{3}{4} = \frac{2}{5}.\]Step 2: Likelihood of one or more selections
The probability that at least one individual is chosen is:\[P({At least one selected}) = 1 - P({No one selected}) = 1 - \frac{2}{5} = \frac{3}{5}.\]Final Result: The probability that at least one individual is chosen is \( \frac{3}{5} \).

Answer 2

Step 1: Compute the probability
The probability \( P(G \cap \overline{H}) \) is calculated as:\[P(G \cap \overline{H}) = P(G) \cdot P(\overline{H}),\]with \( P(G) = \frac{1}{3} \) and \( P(\overline{H}) = 1 - P(H) = 1 - \frac{1}{5} = \frac{4}{5} \).\[P(G \cap \overline{H}) = \frac{1}{3} \cdot \frac{4}{5} = \frac{4}{15}.\]Final Result: The probability \( P(G \cap \overline{H}) \) is \( \frac{4}{15} \).

Answer 3

Step 1: Calculate the probability of precisely one selection
The probability of precisely one selection is determined by the sum of three mutually exclusive scenarios: selecting only R, only J, or only A. This can be expressed as:\[P(\text{Exactly one selected}) = P(R)P(\overline{J})P(\overline{A}) + P(\overline{R})P(J)P(\overline{A}) + P(\overline{R})P(\overline{J})P(A),\]where the probabilities of the complements are:\[P(\overline{J}) = 1 - P(J) = \frac{2}{3}, \quad P(\overline{A}) = 1 - P(A) = \frac{3}{4}, \quad P(\overline{R}) = 1 - P(R) = \frac{4}{5}.\]Substituting the given probabilities:\[P(\text{Exactly one selected}) = \frac{1}{5} \cdot \frac{2}{3} \cdot \frac{3}{4} + \frac{4}{5} \cdot \frac{1}{3} \cdot \frac{3}{4} + \frac{4}{5} \cdot \frac{2}{3} \cdot \frac{1}{4}.\]Simplifying each term:\[P(\text{Exactly one selected}) = \frac{6}{60} + \frac{12}{60} + \frac{8}{60} = \frac{26}{60} = \frac{13}{30}.\]Final Result: The probability of exactly one individual being selected is \( \frac{13}{30} \).

Answer 4

Step 1: Calculate the probability of precisely two selections
The probability of selecting exactly two is given by the sum of probabilities for each specific combination of two selections. This can be expressed as:\[P({Exactly two selected}) = P(R) \cdot P(J) \cdot P(\overline{A}) + P(R) \cdot P(\overline{J}) \cdot P(A) + P(\overline{R}) \cdot P(J) \cdot P(A).\]Substituting the given values:\[P({Exactly two selected}) = \frac{1}{5} \cdot \frac{1}{3} \cdot \frac{3}{4} + \frac{1}{5} \cdot \frac{2}{3} \cdot \frac{1}{4} + \frac{4}{5} \cdot \frac{1}{3} \cdot \frac{1}{4}.\]Simplifying each term and summing them:\[P({Exactly two selected}) = \frac{3}{60} + \frac{2}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20}.\]Final Result: The probability of exactly two individuals being selected is \( \frac{3}{20} \).