Question

Let \(E, F, G\) be three events. Find expressions for the events that of \(E, F, G\) (a) only \(F\) occurs, (b) both \(E\) and \(F\) but not \(G\) occur, (c) at least one event occurs, (d) at least two events occur, (e) all three events occur, (f) none occurs, (g) at most one occurs, (h) at most two occur.

Short answer

\( (a) F \cap E^{c} \cap G^{c}; \newline (b) E \cap F \cap G^{c}; \newline (c) E \cup F \cup G; \newline (d) (E \cap F) \cup (E \cap G) \cup (F \cap G); \newline (e) E \cap F \cap G; \newline (f) E^{c} \cap F^{c} \cap G^{c}; \newline (g) (E \cap F^{c} \cap G^{c}) \cup (E^{c} \cap F \cap G^{c}) \cup (E^{c} \cap F^{c} \cap G) \cup (E^{c} \cap F^{c} \cap G^{c}); \newline (h) (E \cap F \cap G^{c}) \cup (E \cap F^{c} \cap G) \cup (E^{c} \cap F \cap G) \cup (E \cap F^{c} \cap G^{c}) \cup (E^{c} \cap F \cap G^{c}) \cup (E^{c} \cap F^{c} \cap G) \)

Step-by-step solution

Exercise (a) - only F occurs

An expression for the event that only F occurs (and neither E nor G) is: \[ F \cap E^{c} \cap G^{c} \]

Exercise (b) - both E and F but not G occur

An expression where both events E and F occur, and G does not occur, is: \[ E \cap F \cap G^{c} \]

Exercise (c) - at least one event occurs

An expression for the event that at least one of the events E, F, or G occurs is: \[ E \cup F \cup G \]

Exercise (d) - at least two events occur

An expression for the event that at least two of the events E, F, or G occur is: \[ (E \cap F) \cup (E \cap G) \cup (F \cap G) \]

Exercise (e) - all three events occur

An expression for the event that all three events E, F, and G occur is: \[ E \cap F \cap G \]

Exercise (f) - none occurs

An expression for the event that none of the events E, F, or G occurs is: \[ E^{c} \cap F^{c} \cap G^{c} \]

Exercise (g) - at most one occurs

An expression for the event that "at most one" of the events E, F, or G occurs is: \[ (E \cap F^{c} \cap G^{c}) \cup (E^{c} \cap F \cap G^{c}) \cup (E^{c} \cap F^{c} \cap G) \cup (E^{c} \cap F^{c} \cap G^{c}) \]

Exercise (h) - at most two occur

An expression for the event that "at most two" of the events E, F, or G occur is: \[ (E \cap F \cap G^{c}) \cup (E \cap F^{c} \cap G) \cup (E^{c} \cap F \cap G) \cup (E \cap F^{c} \cap G^{c}) \cup (E^{c} \cap F \cap G^{c}) \cup (E^{c} \cap F^{c} \cap G)\]