Use the Poisson Distribution to find indicated probability.
A naturalist leads whale watch trips every morning in March. Teh number of whales seen ha a Poisson distributyion with a mena of 2.2. Find the probability that on a randomly selected trip, the number of whales seen is 5.
Jen:
If you went to the wikipedia site suggested by SraJMcGin, you would have the formula for calculating probabilities following a poisson distribution.
The Poisson distrbution is
f(k,a) =(a^k)*(e^-a)/k! where a is the expected observed in a time period, and k is the actual observed, and e is the exponential e. (Go to the wikipedia site; they can express formulas much better than here on Jiskha.)
In your example a=2.2, and k=5. Plug these into your formula. Hint: I get about 4.8%
Why did the whale go on a trip? Because it wanted to have a whale of a time, of course! Now, to answer your question, we can use the Poisson distribution.
The formula for the Poisson distribution is P(x; μ) = (e^-μ) * (μ^x) / x!, where μ is the mean and x is the number of events we're looking for.
In this case, the mean (μ) is given as 2.2. We want to find the probability of seeing 5 whales (x = 5). Plugging these values into the formula, we get:
P(5; 2.2) = (e^-2.2) * (2.2^5) / 5!
Calculating this out, we find that P(5; 2.2) ≈ 0.1409.
So, the probability of seeing exactly 5 whales on a randomly selected trip is approximately 0.1409, or about 14.09%. Whale done!
To find the probability that on a randomly selected trip, the number of whales seen is 5, we can use the Poisson distribution formula.
The formula for the Poisson distribution is:
P(X = k) = (e^-λ * λ^k) / k!
Where:
- P(X = k) is the probability of exactly k events occurring
- e is the base of the natural logarithm (approximately 2.71828)
- λ is the mean or average number of events occurring in a given time interval
- k is the random variable, or the number of events we are interested in
In this case, the mean (λ) is given as 2.2, and we want to find the probability of seeing exactly 5 whales (k = 5). Thus, we can substitute these values into the formula:
P(X = 5) = (e^-2.2 * 2.2^5) / 5!
Simplifying further, we get:
P(X = 5) = (0.406 * 45.65) / 120
Now, we can calculate the probability:
P(X = 5) = 18.5373 / 120
P(X = 5) ≈ 0.1545
Therefore, the probability that, on a randomly selected trip, the number of whales seen is 5 is approximately 0.1545 or 15.45%.
To find the probability that on a randomly selected trip the number of whales seen is 5, you can use the Poisson distribution formula. The Poisson distribution is given by the formula:
P(x; λ) = (e^-λ * λ^x) / x!
Where:
- P(x; λ) represents the probability of x events occurring in a given interval with an average rate of λ.
- e is the base of the natural logarithm, approximately equal to 2.71828.
- λ is the average rate of events occurring in the given interval.
- x is the number of events you are interested in.
In this case, the average number of whales seen in a trip (λ) is 2.2, and we want to find the probability of seeing 5 whales (x = 5).
Let's substitute these values into the formula:
P(5; 2.2) = (e^-2.2 * 2.2^5) / 5!
Calculating this equation will give us the answer.
Alternatively, you can use statistical software or a scientific calculator with a Poisson distribution function to find the probability directly. Simply input the values λ = 2.2 and x = 5 to get the result.