A marketing company runs a web campaign that includes running an ad on the common platforms. There are an average of 6.1 ad clicks per hour.Use the Poisson distribution to find the probability that there are exactly 5 ad clicks in a given hour.Do not round intermediate computations, and round your answer to three decimal places.(If necessary, consult a list of formulas.)
Question
A marketing company runs a web campaign that includes running an ad on the common platforms. There are an average of 6.1 ad clicks per hour.Use the Poisson distribution to find the probability that there are exactly 5 ad clicks in a given hour.Do not round intermediate computations, and round your answer to three decimal places.(If necessary, consult a list of formulas.)
Solution
The Poisson distribution is a probability distribution that can be used to model the number of events (in this case, ad clicks) in a fixed interval of time. The formula for the Poisson distribution is:
P(X=k) = (λ^k * e^-λ) / k!
where:
- P(X=k) is the probability of k events in the interval,
- λ is the average rate of value (in this case, 6.1 ad clicks per hour),
- e is the base of the natural logarithm (approximately equal to 2.71828),
- k! is the factorial of k.
We want to find the probability that there are exactly 5 ad clicks in a given hour, so we will plug in λ=6.1 and k=5 into the formula:
P(X=5) = (6.1^5 * e^-6.1) / 5!
Now, we just need to calculate this expression.
First, calculate 6.1^5 = 4,572.2961.
Next, calculate e^-6.1 = 0.002478752.
Then, calculate the factorial of 5, which is 54321 = 120.
Finally, multiply and divide these values to get the final probability:
P(X=5) = (4,572.2961 * 0.002478752) / 120 = 0.094.
So, the probability that there are exactly 5 ad clicks in a given hour is approximately 0.094, or 9.4% when rounded to three decimal places.
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