Find the probability when
A. k = 2 and ë = 0.01 for Poisson distribution
B. p = 0.01 and k = 2 for geometric distribution
C. Repeat (b) when binomial distribution is used and n = 10
A. Well, aren't we trying to get poisson-al with numbers here? The probability when k = 2 and ë = 0.01 for the Poisson distribution is given by the formula P(k, ë) = (ë^k * e^(-ë)) / k!, where ë represents the average number of events and k is the number of events we're interested in. Plugging in the values, we get P(2, 0.01) = (0.01^2 * e^(-0.01)) / 2!. Calculate that baby and you've got your probability!
B. Ah, the geometric distribution! It's like waiting for a unicorn to show up at your office party. The probability when p = 0.01 and k = 2 is given by P(k) = (1 - p)^(k-1) * p. So just plug in the values and you'll be one step closer to finding that magical creature, I mean, probability!
C. Let's give the binomial distribution a shot, shall we? With n = 10, p = 0.01, and k = 2, the probability is given by P(k) = C(n, k) * p^k * (1 - p)^(n-k), where C(n, k) represents the binomial coefficient. Just pop in the numbers and you're on your way to finding the probability. And remember, probabilities are like clowns, they can be a bit unpredictable but always ready to clown around!
A. To find the probability when k = 2 and λ = 0.01 for the Poisson distribution, we can use the formula:
P(X = k) = (e^(-λ) * λ^k) / k!
In this case, k = 2 and λ = 0.01. We can plug these values into the formula to find the probability:
P(X = 2) = (e^(-0.01) * 0.01^2) / 2!
To calculate this, you need to use the value of e (approximately 2.71828) and find its negative exponent value using a scientific calculator. Then, multiply it by 0.01 squared (0.0001) and divide by 2 factorial (2 * 1 = 2). This will give you the probability when k equals 2 and λ equals 0.01 for the Poisson distribution.
B. To find the probability when p = 0.01 and k = 2 for the geometric distribution, we can use the formula:
P(X = k) = (1-p)^(k-1) * p
In this case, p = 0.01 and k = 2. We can plug these values into the formula to find the probability:
P(X = 2) = (1-0.01)^(2-1) * 0.01
To calculate this, subtract p from 1 (1-0.01 = 0.99), raise it to the power of k-1 (0.99^1 = 0.99), and multiply it by p (0.99 * 0.01 = 0.0099). This will give you the probability when p equals 0.01 and k equals 2 for the geometric distribution.
C. To repeat part B, but using the binomial distribution and n = 10, we can use the formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
In this case, p = 0.01, k = 2, and n = 10. We can plug these values into the formula to find the probability:
P(X = 2) = (10 choose 2) * 0.01^2 * (1-0.01)^(10-2)
To calculate this, you need to evaluate the binomial coefficient (10 choose 2) which is equal to 45, raise p to the power of k (0.01^2 = 0.0001), raise (1-p) to the power of (n-k) (0.99^8 = approximately 0.922), and multiply all these values (45 * 0.0001 * 0.922). This will give you the probability when p equals 0.01, k equals 2, and n equals 10 for the binomial distribution.