A toy manufacturing company that 5% of there toys are defective. A quality control person checks 50 toys.
a) What is the probability that there will be exactly two defective toys found?
b) How many defective toys would quality control expect to find?
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To find the answers to these questions, we can use the binomial probability distribution formula. The binomial distribution is used when we have a fixed number of trials (in this case, checking 50 toys) and each trial has two possible outcomes (defective or non-defective).
The formula for the probability of getting exactly "k" successes in "n" trials is:
P(X=k) = (nCk) * p^k * (1-p)^(n-k)
Where:
P(X=k) is the probability of getting exactly "k" successes (defective toys)
n is the total number of trials (toys checked)
k is the number of successful outcomes (defective toys)
p is the probability of having a defective toy in each trial
1-p is the probability of not having a defective toy in each trial
nCk is the combination formula "n choose k"
a) Probability of finding exactly two defective toys:
Using the given information, we know that p = 0.05 (5% defective toys). We also know that n = 50 (total toys checked), and k = 2 (number of defective toys).
Plugging these values into the formula, we can calculate the probability as follows:
P(X=2) = (50C2) * (0.05^2) * (1-0.05)^(50-2)
You can use a calculator or a program (like Python or Excel) to calculate the combination and the powers. In this case, (50C2) = 1225. Substituting this value, we get:
P(X=2) = 1225 * (0.05^2) * (0.95^48)
Using a calculator, the result is approximately 0.2416, or 24.16%.
Therefore, the probability of finding exactly two defective toys is approximately 24.16%.
b) Expected number of defective toys:
To find the expected number of defective toys, we need to multiply the total number of toys checked (n) by the probability of having a defective toy in each trial (p).
Expected number = n * p
Expected number = 50 * 0.05
Expected number = 2.5
Therefore, quality control would expect to find approximately 2.5 defective toys.