Suppose that past history shows that 20% of Sydney trains are late. A sample of 12 trains is selected for an experiment.The probability that at most 4 trains is late is Question 12Select one:a.0.800b. 0.133c. 0.587d. 0.928
Question
Suppose that past history shows that 20% of Sydney trains are late. A sample of 12 trains is selected for an experiment.The probability that at most 4 trains is late is Question 12Select one:a.0.800b. 0.133c. 0.587d. 0.928
Solution
This is a binomial distribution problem. The formula for binomial distribution is:
P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))
where:
- P(X=k) is the probability of k successes in n trials
- C(n, k) is the combination of n items taken k at a time
- p is the probability of success on a single trial
- n is the number of trials
- k is the number of successes
Given in the problem:
- p = 0.20 (probability that a train is late)
- n = 12 (number of trains selected)
- k = 0, 1, 2, 3, 4 (we want the probability that at most 4 trains are late, so we need to calculate for each of these and sum them up)
We can use a binomial calculator to find these probabilities or calculate manually.
P(X=0) = C(12, 0) * (0.20^0) * ((1-0.20)^(12-0)) P(X=1) = C(12, 1) * (0.20^1) * ((1-0.20)^(12-1)) P(X=2) = C(12, 2) * (0.20^2) * ((1-0.20)^(12-2)) P(X=3) = C(12, 3) * (0.20^3) * ((1-0.20)^(12-3)) P(X=4) = C(12, 4) * (0.20^4) * ((1-0.20)^(12-4))
Adding these probabilities together will give us the probability that at most 4 trains are late.
If you calculate these values, you will find that the sum is approximately 0.928. So, the correct answer is d. 0.928.
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