We are given the probabilities of events $A$, $B$, $C$, $AB$ (which means $A \cap B$), $AC$ (which means $A \cap C$), $BC$ (which means $B \cap C$), and $ABC$ (which means $A \cap B \cap C$). We are asked to find the probabilities of $A \cup B$, $\overline{AB}$ (complement of $A \cap B$), $A \cup B \cup C$, $\overline{ABC}$ (complement of $A \cap B \cap C$).

1. Problem Description

We are given the probabilities of events $A$, $B$, $C$, $AB$ (which means $A \cap B$), $AC$ (which means $A \cap C$), $BC$ (which means $B \cap C$), and $ABC$ (which means $A \cap B \cap C$). We are asked to find the probabilities of $A \cup B$, $\overline{AB}$ (complement of $A \cap B$), $A \cup B \cup C$, $\overline{ABC}$ (complement of $A \cap B \cap C$).

2. Solution Steps

a. $P(A \cup B)$:

We use the formula:

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

$P(A) = \frac{1}{2}$

$P(B) = \frac{1}{3}$

$P(A \cap B) = P(AB) = \frac{1}{10}$

$P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{10} = \frac{15}{30} + \frac{10}{30} - \frac{3}{30} = \frac{25-3}{30} = \frac{22}{30} = \frac{11}{15}$

b. $P(\overline{AB})$:

We use the formula:

$P(\overline{AB}) = 1 - P(AB)$

$P(AB) = \frac{1}{10}$

$P(\overline{AB}) = 1 - \frac{1}{10} = \frac{10}{10} - \frac{1}{10} = \frac{9}{10}$

c. $P(A \cup B \cup C)$:

We use the formula:

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

$P(A) = \frac{1}{2}$

$P(B) = \frac{1}{3}$

$P(C) = \frac{1}{5}$

$P(A \cap B) = \frac{1}{10}$

$P(A \cap C) = \frac{1}{15}$

$P(B \cap C) = \frac{1}{20}$

$P(A \cap B \cap C) = \frac{1}{30}$

$P(A \cup B \cup C) = \frac{1}{2} + \frac{1}{3} + \frac{1}{5} - \frac{1}{10} - \frac{1}{15} - \frac{1}{20} + \frac{1}{30}$

$P(A \cup B \cup C) = \frac{30}{60} + \frac{20}{60} + \frac{12}{60} - \frac{6}{60} - \frac{4}{60} - \frac{3}{60} + \frac{2}{60}$

$P(A \cup B \cup C) = \frac{30 + 20 + 12 - 6 - 4 - 3 + 2}{60} = \frac{64 - 13 + 2}{60} = \frac{51 + 2}{60} = \frac{53}{60}$

d. $P(\overline{ABC})$:

We use the formula:

$P(\overline{ABC}) = 1 - P(ABC)$

$P(ABC) = \frac{1}{30}$

$P(\overline{ABC}) = 1 - \frac{1}{30} = \frac{30}{30} - \frac{1}{30} = \frac{29}{30}$

3. Final Answer

$P(A \cup B) = \frac{11}{15}$

$P(\overline{AB}) = \frac{9}{10}$

$P(A \cup B \cup C) = \frac{53}{60}$

$P(\overline{ABC}) = \frac{29}{30}$