Final answer:
The probability that the celebrity posts no messages on any random day is approximately 0.111, calculated using the Poisson distribution formula with λ equal to 2.2.
Step-by-step explanation:
To find the probability that a celebrity posts no messages on a social media site, given that their average number of posts per day is 2.2, we can model this with a Poisson distribution. In a Poisson distribution, the probability of observing exactly k events in a given interval is given by:
P(X=k) = (λ^k * e^-λ) / k!
Where λ is the average number of events (in our case, messages per day), k is the number of events we want to find the probability for, and e is the base of the natural logarithm (approximately 2.71828).
For this question, we want to find the probability of k = 0 events, so our formula simplifies to:
P(X=0) = (2.2^0 * e^-2.2) / 0!
Which simplifies to:
P(X=0) = (1 * e^-2.2) / 1
P(X=0) = e^-2.2
P(X=0) ≈ 0.1108 (rounded to four decimal places)
Therefore, the probability that the celebrity does not post any messages on any random day is approximately 0.111 (rounded to three decimal places).