A company makes microphone cable for recording studios. They have been finding an average of 24 flaws for every 40 meters of cable.Use the Poisson distribution to find the probability that there are exactly 0 flaws in a given meter of cable.Do not round intermediate computations, and round your answer to three decimal places.(If necessary, consult a list of formulas.)
Question
A company makes microphone cable for recording studios. They have been finding an average of 24 flaws for every 40 meters of cable.Use the Poisson distribution to find the probability that there are exactly 0 flaws in a given meter of cable.Do not round intermediate computations, and round your answer to three decimal places.(If necessary, consult a list of formulas.)
Solution
To solve this problem, we first need to find the average number of flaws per meter. We know that there are 24 flaws for every 40 meters, so the average number of flaws per meter (λ) is 24/40 = 0.6 flaws per meter.
The Poisson distribution formula is:
P(x; λ) = (e^-λ * λ^x) / x!
where:
- P(x; λ) is the probability of x events in an interval,
- e is the base of the natural logarithm (approximately equal to 2.71828),
- λ is the average rate of value,
- x is the actual number of successes that result from the experiment.
We want to find the probability of exactly 0 flaws in a given meter of cable, so x = 0.
Substituting these values into the formula, we get:
P(0; 0.6) = (e^-0.6 * 0.6^0) / 0!
Solving this, we get:
P(0; 0.6) = (0.54881 * 1) / 1 = 0.54881
So, the probability that there are exactly 0 flaws in a given meter of cable is approximately 0.549 (rounded to three decimal places).
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