Question

^{QUESTION 14 The number of flaws in a given area of aluminum folil is 3 per m2 Find the probability that there are no flaws in one-square meter of aluminum foil? Round your answer to 3 decimal places. 0.050 QUESTION 15 The number of flaws in a given area of aluminum foil is 3 per m2. Find the probability that there are more than 3 flaws in one-square meter of aluminum foil? Round your answer to 3 decimal places. 0.353 QUESTION 16 The number of flaws in a given area of aluminum foil is 3 per m2. What is the expected number of flaws in TWO square meters of aluminum foil? Round your answer to be an integer. QUESTION 17 The number of flaws in a given area of aluminum foil is 3 per m2 Find the probability that there are at least 8 flaws in TWO square meters of aluminum foil? Round your answer to 3 decimal places}

Answer 1

Since the average number of flaws that occurs in a specified region (area) is known and the probability that a flaw will occur is proportional to the size of the region (area), the number of flaws in a given area can be modeled as Poisson distribution.

Let X be the number of flaws in 1 m²

Then X ~ Poisson($\lambda$ = 3)

14.

Probability of no flaw = P(X = 0) = exp(-3) * 3⁰ / 0!

= exp(-3)

= 0.05

15.

Probability of more than 3 flaws = P(X > 3)

= 1 - P(X $\le$ 3)

= 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]

= 1 - [exp(-3) * 3⁰ / 0! + exp(-3) * 3¹ / 1! + exp(-3) * 3² / 2! + exp(-3) * 3³ / 3! ]

= 1 - (0.04978707 + 0.14936121 + 0.22404181 + 0.22404181)

= 0.353

16.

Rate $\lambda$ for 1 square meter = 3

Rate $\lambda$ for 2 square meters = 3 * 2 = 6

17.

For 2 square meters, $\lambda$ = 6

Probability that there are at least 8 flaws = P(X $\ge$ 8)

= 1 - P(X < 8)

= 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)]

= 1 - [exp(-6) * 6⁰ / 0! + exp(-6) * 6¹ / 1! + exp(-6) * 6² / 2! + exp(-6) * 6³ / 3! + exp(-6) * 6⁴ / 4! + exp(-6) * 6⁵ / 5! + exp(-6) * 6⁶ / 6! + exp(-6) * 6⁷ / 7!]

= 1 - [0.002478752 + 0.014872513 + 0.044617539 + 0.089235078 + 0.133852618 + 0.160623141 + 0.160623141 + 0.137676978]

= 0.256