Question 12.11E2: Quality Control for Batteries A manufacturer of batteries kn......
Quality Control for Batteries
A manufacturer of batteries knows that 0.4% of the batteries produced by the company are defective.
a) Write the binomial probability formula that would be used to determine the probability that exactly x out of n batteries produced are defective.
b) Write the binomial probability formula that would be used to find the probability that exactly 3 batteries of 75 produced will be defective. Do not evaluate.
a) We want to find the probability that exactly x batteries are defective where selecting a defective battery is considered success. The probability, p, that an individual battery is defective is 0.4%, or 0.004 in decimal form. The probability that a battery is not defective, q, is 1 – 0.004, or 0.996. The general formula for finding the probability that exactly x out of n batteries produced are defective is
P(x) = (_nC_x)p^xq^{n-x}
Substituting 0.004 for p and 0.996 for q, we obtain the formula
P(x) = (_nC_x)(0.004)^x(0.996)^{n-x}
b) We want to determine the probability that exactly 3 batteries out of 75 produced are defective. Thus, x = 3 and n = 75. Substituting these values into the formula in part (a) gives
P(3) = (_{75}C_3)(0.004)^3(0.996)^{75-3}
= (_{75}C_3)(0.004)^3(0.996)^{72}
The answer may be obtained using a scientific calculator.