Asked by Lucille Nicole on May 09, 2024
Verified
The probability that a car will have a flat tire while driving through a certain tunnel is 0.00006.Use the Poisson approximation to the binomial distribution to find the probability that among 13,000 cars passing through this tunnel,at most two will have a flat tire.
A) 0.1394
B) 0.1840
C) 0.9554
D) 0.8160
E) 0.8606
Poisson Approximation
A statistical technique used to estimate the probability of a given number of events happening in a fixed interval of time or space when these events occur with a known constant mean rate and independently of the time since the last event.
Probability
An indicator representing the probability or odds of a specific event happening, with its value ranging from 0 to 1.
- Engage the Poisson approximation for binomial distribution calculations in probability assessments.
Verified Answer
CR
Christine RemickMay 12, 2024
Final Answer :
C
Explanation :
The Poisson approximation to the binomial distribution can be used with λ = np, where n is the number of trials (13,000 cars) and p is the probability of success (0.00006). Thus, λ = 13,000 * 0.00006 = 0.78. The probability of at most 2 cars having a flat tire is given by P(X ≤ 2) = P(0) + P(1) + P(2), where P(k) = (e^(-λ) * λ^k) / k!. Substituting λ = 0.78, we calculate P(0), P(1), and P(2), and sum them up to find the probability, which is approximately 0.9554.
Learning Objectives
- Engage the Poisson approximation for binomial distribution calculations in probability assessments.