Question 1186890: Let A, B and C be three arbitrary events. Find expressions for the following events that
of A, B and C.
a- Only A occurs, b- All three events occur, c- Non occurs,
d- At least one occurs, e- Not more than 2 occur, g-One and only one occurs.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here are the expressions for the given events using set notation:
**a. Only A occurs:**
A ∩ Bᶜ ∩ Cᶜ (A intersects with the complements of B and C)
**b. All three events occur:**
A ∩ B ∩ C (The intersection of all three events)
**c. None occurs:**
Aᶜ ∩ Bᶜ ∩ Cᶜ (The intersection of the complements of all three events). This can also be written as (A ∪ B ∪ C)ᶜ (De Morgan's Law).
**d. At least one occurs:**
A ∪ B ∪ C (The union of all three events)
**e. Not more than 2 occur:**
This is equivalent to "none occur" or "exactly one occurs" or "exactly two occur". It is the complement of "all three occur".
(A ∩ B ∩ C)ᶜ
Alternatively, we can express it as the union of the cases where no events occur, only one occurs, and exactly two occur.
(Aᶜ ∩ Bᶜ ∩ Cᶜ) ∪ (A ∩ Bᶜ ∩ Cᶜ) ∪ (Aᶜ ∩ B ∩ Cᶜ) ∪ (Aᶜ ∩ Bᶜ ∩ C) ∪ (A ∩ B ∩ Cᶜ) ∪ (A ∩ Bᶜ ∩ C) ∪ (Aᶜ ∩ B ∩ C)
**f. One and only one occurs:**
(A ∩ Bᶜ ∩ Cᶜ) ∪ (Aᶜ ∩ B ∩ Cᶜ) ∪ (Aᶜ ∩ Bᶜ ∩ C) (The union of the three mutually exclusive scenarios where only one of the events occurs).
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