MATH SOLVE

5 months ago

Q:
# Of the tools produced with a certain manufacturing process, 10% are defective. Using: a) the binomial distribution and b) the Poisson approximation to the binomial distribution, find the probability that in a sample of 10 tools chosen at random exactly 2 are defective.

Accepted Solution

A:

a) Using the Binomial Distribution:
In this case, we have a binomial distribution with the following parameters:
Number of trials (n): 10 (sample size).
Probability of success (p) for each trial (defective tool): 0.10 (10% are defective).
Probability of failure (q) for each trial (non-defective tool): 1 - 0.10 = 0.90.
We want to find the probability of getting exactly 2 defective tools, which can be calculated using the binomial probability formula:
$$P(X=k)=nC_k\cdot{p^k}\cdot{q^{n−k}}$$
Where:
P(X=k) is the probability of getting exactly k defective tools.
n is the number of trials (10).
k is the number of successes (2).
p is the probability of success (0.10).
q is the probability of failure (0.90).
Now, plug in the values and calculate:
$$P(X=2)=10C2\cdot(0.10)^2(0.90)^8$$
Using the binomial coefficient
$$P(X=2)=45\cdot(0.10)^2(0.90)^8$$
Calculate the result:
P(X=2)=0.1937
So, using the binomial distribution, the probability of getting exactly 2 defective tools in a sample of 10 is approximately 0.1937.