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: (Use the poisson approximation to the binomial.)
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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: 1. None of the antennas is defective.
To find the probability in this scenario, we can use the Poisson approximation to the binomial distribution. The steps to find the probability are as follows:
Step 1: Identify the parameters:
In this problem, the parameter is the probability (p) of a defective antenna, which is given as 1.5 percent or 0.015. The sample size (n) is also given as 200.
Step 2: Calculate the mean (λ):
The mean (λ) is calculated by multiplying the sample size (n) by the probability (p).
λ = n * p = 200 * 0.015 = 3
Step 3: Apply the Poisson approximation:
The Poisson distribution can be used as an approximation to the binomial distribution when the sample size is large and the probability of success is small. In this case, since λ = 3 is relatively small, the Poisson approximation can be applied.
Step 4: Calculate the desired probability using the Poisson distribution formula:
The probability of obtaining a specific number of defective antennas in the sample can be calculated using the Poisson distribution formula:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
X = number of defective antennas
k = desired number of defective antennas
λ = mean
In this problem, we need to find the probability of having a specific number of defective antennas in the sample, so we can substitute the values into the formula.
P(X = 0) = (e^(-3) * 3^0) / 0!
P(X = 1) = (e^(-3) * 3^1) / 1!
P(X = 2) = (e^(-3) * 3^2) / 2!
...
P(X = 200) = (e^(-3) * 3^200) / 200!
To find the desired probability, you would need to calculate the individual probabilities for the desired number of defective antennas and then sum them up.