As a result of an experiment one of four mutually exclusive…
As a result of an experiment one of four mutually exclusive events can occur, with probabilities $0.012$, $0.010$, $0.006$ and $0.002$ respectively. Determine the probability that one of these events occurs as a result of the experiment.
Explanation of Method
For mutually exclusive (incompatible) events the probability that one of them occurs equals the sum of their probabilities — the addition rule. We note why the events must be incompatible for this to work.
Step 1: The addition rule
For pairwise incompatible events $A_1,\dots,A_4$:
\[ P(A_1+A_2+A_3+A_4)=P(A_1)+P(A_2)+P(A_3)+P(A_4). \]
Step 2: The answer
\[ \boxed{P=0.012+0.010+0.006+0.002=0.030.} \]
Step 3: Remark
If the events could occur together, the sum $0.030$ would only be an upper bound (Boole's inequality), and the exact answer would require the probabilities of their intersections. For example, for four independent events the answer would be $1-0.988\cdot0.990\cdot0.994\cdot0.998\approx0.0297$ — slightly less than $0.03$.
Result
| Answer | |
|---|---|
| Probability | $0.012+0.010+0.006+0.002=0.03$ |