A naturalist leads whale watch trips every morning in March. The number of whales seen has a Poisson distribution with a mean of 2.1 Find the probability that on a randomly selected trip, the number of whales seen is 3. Round to four decimal places.
Why did the whale bring a calculator on the trip? Because it wanted to calculate its chances of being spotted!
But to answer your question seriously, we can use the Poisson distribution formula to calculate the probability. The formula is:
P(x, λ) = (e^(-λ) * λ^x) / x!
Where P(x, λ) is the probability of seeing x whales given an average of λ whales.
In your case, λ is 2.1 and x is 3. Plugging the values into the formula:
P(3, 2.1) = (e^(-2.1) * 2.1^3) / 3!
Calculating this gives us approximately 0.2297. So, the probability of seeing 3 whales on a randomly selected trip is approximately 0.2297.
To calculate the probability that on a randomly selected trip, the number of whales seen is 3, we will 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 seeing x events occur
λ is the average number of events that occur in a specific time interval
e is the mathematical constant approximately equal to 2.71828
x is the number of events we are interested in finding the probability for
In this case, the average number of whales seen on a trip is given as λ = 2.1. We want to find the probability of seeing exactly 3 whales, so x = 3.
Using the formula, we can calculate the probability as follows:
P(3; 2.1) = (e^(-2.1) * 2.1^3) / 3!
Calculating this expression:
P(3; 2.1) ≈ (2.71828^(-2.1) * 2.1^3) / 6
P(3; 2.1) ≈ (0.122092 * 9.261) / 6
P(3; 2.1) ≈ 0.183
Rounded to four decimal places, the probability that on a randomly selected trip, the number of whales seen is 3 is approximately 0.183.
To find the probability of seeing 3 whales on a randomly selected trip, we can use the Poisson distribution formula. The formula for the probability of a Poisson random variable is:
P(x; μ) = (e^(-μ) * μ^x) / x!
Where:
- P(x; μ) is the probability of observing x events
- e is the mathematical constant approximately equal to 2.71828
- μ is the mean number of events in the given interval
- x is the actual number of events observed
In this case, the mean number of whales seen on a trip is 2.1. We want to find the probability of observing 3 whales, so we substitute these values into the formula:
P(x = 3; μ = 2.1) = (e^(-2.1) * 2.1^3) / 3!
Using a calculator, we find that e^(-2.1) ≈ 0.1224 and 3! = 3 * 2 * 1 = 6. Substituting these values, we get:
P(x = 3; μ = 2.1) = (0.1224 * 2.1^3) / 6
Evaluating 2.1^3 ≈ 9.261, we can further simplify the equation:
P(x = 3; μ = 2.1) ≈ (0.1224 * 9.261) / 6
Calculating the numerator, we get:
P(x = 3; μ = 2.1) ≈ 1.1334 / 6
And finally, dividing 1.1334 by 6, we find:
P(x = 3; μ = 2.1) ≈ 0.1889
Hence, the probability of observing 3 whales on a randomly selected trip is approximately 0.1889.