On the average five customers visit a children department store each hour. Using Poisson distribution, calculate the probability of three customers shop at the store.
Poisson distribution (m = mean):
P(x) = e^(-m) m^x / x!
Values:
x = 3
m = 5
Substitute and calculate.
Well, let's put on our mathematical clown noses and have some fun! To find the probability of three customers visiting the store, we'll use the Poisson distribution.
The Poisson distribution formula is P(x; λ) = (e^(-λ) * λ^x) / x!, where λ is the average rate of occurrence (in this case, five customers per hour), x is the number of occurrences we want (three customers), e is Euler's number (approximately 2.71828), and ! represents the factorial operation.
Plugging in the values, we get:
P(3; 5) = (e^(-5) * 5^3) / 3!
Now, let me calculate this for you...
[Calculating in silence...]
[Drumroll...]
[confetti]
According to my calculations, the probability of exactly three customers shopping at the store in an hour is approximately 0.10082, or around 10.08%.
So, there you have it! The probability of three customers shopping at the store is quite a decent chance. Just remember, even in a children's store, real-life clowns love probabilities and math!
To calculate the probability of three customers shopping at the store, we can use the Poisson distribution formula.
The Poisson distribution formula is given by:
P(x; λ) = (e^(-λ) * λ^x) / x!
Where:
- P(x; λ) is the probability of getting exactly x occurrences in a given time period,
- e is the base of the natural logarithm (approximately 2.71828),
- λ (lambda) is the average rate of occurrence in the given time period, and
- x is the actual number of occurrences we are interested in.
In this case, we are given that the average rate of customers visiting the store is five customers per hour.
Using this information, we can plug in the values into the Poisson distribution formula:
P(3; 5) = (e^(-5) * 5^3) / 3!
Now, we can calculate this probability step-by-step:
1. Calculate e^(-5):
e^(-5) = 0.006737946999085467
2. Calculate 5^3:
5^3 = 125
3. Calculate 3! (3 factorial):
3! = 3 * 2 * 1 = 6
4. Substitute these values into the formula:
P(3; 5) = (0.006737946999085467 * 125) / 6
5. Calculate this expression:
P(3; 5) ≈ 0.1403738958142807
Therefore, the probability of three customers shopping at the store is approximately 0.1404 (or 14.04%).
To calculate the probability using the Poisson distribution, we can use the formula:
P(x; λ) = e^-λ * (λ^x) / x!
where:
P(x; λ) is the probability of x events occurring given an average rate of λ,
e is Euler's number (approximately 2.71828),
^ denotes exponentiation,
λ is the average rate of events over a given time period, and
x is the number of events we are interested in.
In this case, the average rate λ is given as five customers per hour. We are interested in the probability of three customers shopping, so x is three.
Let's calculate it step by step:
Step 1: Calculate the value of λ:
λ = 5 customers/hour
Step 2: Plug the values into the Poisson distribution formula:
P(3; 5) = e^-5 * (5^3) / 3!
Step 3: Evaluate the formula:
P(3; 5) = (2.71828^-5) * (5^3) / (3 * 2 * 1)
Step 4: Calculate the Poisson probability:
P(3; 5) ≈ 0.10081
So, the probability of three customers shopping at the store, based on the Poisson distribution, is approximately 0.10081.