Suppose 1.5 percent of the antennas on new Nokia cell phones are defective. For a random sample of 200 antennas, find the probability that 5 or more will be defective
A = .0498
B = .5768
For this, I would use a Poisson distribution. (There are other methods).
Under a Poisson, the probability of observing exactly k true events is:
P=[y^k * e^(-y)]/k!
where y is the expected number = 3
So, P(0) = [3^0 * e^(-3)/0! = .04979
P(1) = 3^1 * e^(-3)/1! = .14946
and so on.
Take it from here.
To find the probability that 5 or more antennas out of 200 are defective, we can use the binomial probability formula. The formula for the binomial probability is given by
P(X=k) = (n C k) * p^k * (1-p)^(n-k),
where P(X=k) is the probability of getting exactly k successes, n is the number of trials, p is the probability of success, and (n C k) denotes the number of combinations of n items taken k at a time.
In this case, n = 200, p = 0.015 (probability of defective antenna), and we want to find the probability of having 5 or more defective antennas (k ≥ 5).
Let's calculate the probability using this formula step by step:
First, calculate the probability of exactly 5 defective antennas:
P(X=5) = (200 C 5) * (0.015)^5 * (1-0.015)^(200-5)
Using the combination formula (n C k) = n! / (k! * (n-k)!), we have:
P(X=5) = (200! / (5! * 195!)) * (0.015)^5 * (1-0.015)^195
Next, calculate the probability of exactly 6 defective antennas:
P(X=6) = (200 C 6) * (0.015)^6 * (1-0.015)^(200-6)
Continuing this pattern, calculate the probabilities for 7, 8, ..., up to 200 defective antennas.
Finally, sum up all these probabilities to get the total probability of having 5 or more defective antennas:
P(X ≥ 5) = P(X=5) + P(X=6) + ... + P(X=200)
To solve this problem, we can use the binomial probability formula. The binomial probability formula calculates the probability of obtaining a specific number of successes in a fixed number of trials, given the probability of success in each trial.
The formula for calculating the probability of getting exactly k successes in n trials, given a probability of success p, is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of getting exactly k successes
C(n, k) is the number of ways to choose k items from a set of n items, also known as the binomial coefficient, calculated as C(n, k) = n! / (k! * (n - k)!)
p is the probability of success in each trial
k is the number of successes we are interested in
n is the total number of trials
In this case, the probability of an antenna being defective (success) is 1.5 percent, which can be expressed as 0.015. We want to find the probability of having 5 or more defective antennas in a sample of 200 antennas.
Let's calculate the probability using the formula:
P(X >= 5) = P(X = 5) + P(X = 6) + ... + P(X = 200)
P(X = k) = C(200, k) * (0.015)^k * (1 - 0.015)^(200 - k)
Now we need to calculate the probability for each value of k from 5 to 200 and sum them up.
Since calculating this by hand can be quite tedious, let's use a calculator or a statistical software to find the answer more easily.