An online calculator to calculate the Poisson probability distribution and the probabilities of "at least" and "at most" related to the binomials.
Example 1
Event A occurs at an average of 4 times over a period of 24 hours.
a) What is the probability that event A occurs 5 times over a period of 24 hours?
b) What is the probability that event A occurs at most 5 times over a period of 24 hours?
c) What is the probability that event A occurs at least 5 times over a period of 24 hours?
Solution to Example 1
The average \( \lambda = 4 \) is over a period of 24 hours. The probability to be calculated is over the same period. Hence
a)
\( P(X = 5) = \dfrac{e^{-4}4^5}{5!} = 0.15629 \)
b)
At most 5 times means \( x \) is either \( 0, 1, 2 , 3, 4 \; \text{or} \; 5\) or \( x \le 5 \)
\( P(\text{at most 5 times}) = P( X = 0 \; or \; X = 1 \; or \; X = 2 \; \)
\( or \; X = 3 or \; X = 4 \; or \; X = 5 ) \)
Using the binomial formula, the probability may be written as
\( P(X \le 5) = P(X = 0) + P(X = 1) + P(X = 2) \)
\( + P(X = 3) + P(X = 4) + P(X = 5) \)
\( = 0.018315 + 0.073262 + 0.146525 + 0.195366 + 0.195366 + 0.156293 \)
\(= 0.78513\)
c)
At least 5 times means \( x \) is equal or greater than 5.
\( P(\text{at least 5 times}) = 1 - P(\text{at most 4 times}) \)
\( = 1 - ( P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) ) \)
\( = 1 - (0.018315 + 0.073262 + 0.146525 + 0.195366 + 0.195366 ) \)
\( = 0.37116 \)
