A statistics professor receives an average of five e-mail messages per day from students. Assume the number of messages approximates a Poisson distribution. What is the probability that on a randomly selected day she will have five messages?
0.0067
0.0842
To find the probability that the statistics professor will have five messages on a randomly selected day, we can use the Poisson probability formula.
The Poisson probability formula is given by:
P(x; λ) = (e^(-λ) * λ^x) / x!
where:
- P(x; λ) is the probability of getting x occurrences in a given time period
- e is the base of the natural logarithm (approximately 2.71828)
- λ is the average number of occurrences in the given time period
- x is the actual number of occurrences we are interested in
- x! denotes the factorial of x
In this case, the average number of messages received by the professor per day (λ) is 5, and we want to find the probability of having exactly 5 messages (x = 5).
Using the Poisson probability formula, we can substitute the values:
P(5; 5) = (e^(-5) * 5^5) / 5!
Now, let's calculate it step-by-step.
Step 1: Calculate e^(-5)
e^(-5) ≈ 0.00673794699
Step 2: Calculate 5^5
5^5 = 3125
Step 3: Calculate 5!
5! = 5 * 4 * 3 * 2 * 1 = 120
Step 4: Calculate P(5; 5)
P(5; 5) = (0.00673794699 * 3125) / 120 ≈ 0.175467
Therefore, the probability that the statistics professor will have exactly five messages on a randomly selected day is approximately 0.175467 or 17.55%.
To find the probability that the statistics professor will receive exactly five messages on a randomly selected day, we can use the Poisson distribution formula.
The Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time or space, given the average rate of occurrence.
In this case, the professor receives an average of five email messages per day. This average rate of occurrence is denoted by λ (lambda) in the Poisson distribution formula.
The probability mass function (PMF) of the Poisson distribution is given by:
P(X = k) = (e^(-λ) * λ^k) / k!
where:
k - the number of events (in this case, number of messages)
λ - the average rate of occurrence
In our case, we want to find P(X = 5), where X is the number of messages.
Plugging in the values:
λ = 5 (average number of messages per day)
k = 5 (we want to find the probability of exactly 5 messages)
P(X = 5) = (e^(-5) * 5^5) / 5!
Using a calculator or software, you can evaluate this expression to find the probability that the professor will have exactly five messages on a randomly selected day.