At a cafeteria the customers arrive at an average of 0.3 per minute. The probability that a)exactly 2 customers arrive in a 10 minute span b)2 or more customers arrive in a 10 minute span c) one customer arrives in a 5 minute span and one customer arrives in the next minute..
To solve these probability problems, we need to use the Poisson distribution formula. The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given the average rate of occurrence. The Poisson distribution formula is:
P(x; λ) = (e^(-λ) * λ^x) / x!
Where:
- P(x; λ) is the probability of x events occurring in the given interval of time or space.
- e is the mathematical constant approximately equal to 2.71828.
- λ is the average rate of events occurring in the given interval.
- x is the number of events we are interested in.
a) To find the probability that exactly 2 customers arrive in a 10-minute span, we need to calculate P(x = 2; λ = 0.3 * 10) using the Poisson distribution formula.
P(x = 2; λ = 0.3 * 10) = (e^-(0.3 * 10) * (0.3 * 10)^2) / 2!
Calculating this expression will give us the answer.
b) To find the probability that 2 or more customers arrive in a 10-minute span, we need to sum up the probabilities of 2, 3, 4, and so on. Using the Poisson distribution formula, we calculate the probabilities for each value of x (2, 3, 4, and so on) and add them up until we reach a cumulative probability of 1. This sum will give us the answer.
c) To find the probability that one customer arrives in a 5-minute span and another customer arrives in the next minute, we can multiply the probabilities of these two independent events occurring. The probability of one customer arriving in a 5-minute span can be calculated using the Poisson distribution formula P(x = 1; λ = 0.3 * 5), and the probability of another customer arriving in the next minute can be calculated using P(x = 1; λ = 0.3 * 1). Multiplying these probabilities will give us the answer.