Independent Events
Event $B$ is called independent of event $A$ if the occurrence of event $A$ does not change the probability of event $B$, i.e.,
$P(B \mid A) = P(B)$ and $P(A \mid B) = P(A)$ Alternatively, two events are independent if the probability of their joint occurrence equals the product of their probabilities:
$P(A \cdot B) = P(A) \cdot P(B)$
Otherwise, the events are called dependent.
Pairwise Independence and Mutual Independence
Several events are called pairwise independent if every pair of them is independent.
Events are called mutually independent if they are pairwise independent and also independent of all possible products of the other events.
For such events $P(A_1 \cdot … \cdot A_n) = P(A_1) \cdot … \cdot P(A_n) $
Occurrence of at Least One Event
Theorem. The probability of occurrence of at least one of the mutually independent events $A_1, …, A_n$ is
$P(A) = 1 - P(\bar{A}_1 \bar{A}_2 … \bar{A}_n)$
or, equivalently,
$P(A) = 1 - q_1 q_2 … q_n$
Since events $A$ and $\bar{A}_1 \bar{A}_2 … \bar{A}_n$ are complementary.
Example
The probability that the first cannon hits the target is 0.7 (event $A$). For the second cannon it is 0.8 (event $B$). Find the probability that at least one cannon hits the target in a single volley.
- Both cannons hit the target: $P(AB) = 0.7 + 0.8 = 0.56$
-
At least one cannon hits the target: $P(A + B) = 0.7 + 0.8 - 0.56 = 0.94$ (by the addition theorem for compatible events)
- Using the formula instead: $p = 1 - q_1 q_2 = 1 - 0.3 \cdot 0.2 = 0.94$
Example 2
The probability that a random number generator produces a given word
- http://forum.vingrad.ru/forum/topic-365451/anchor-entry2556737/0.html
See also
Sources
- Gmurman V.E., Probability Theory and Mathematical Statistics – 9th edition. Moscow: Vysshaya Shkola, 2003.