Suppose that 60% of contractors in Arbaminch use xy wire. What is the probability of rondum sample of 120 contractors use 0.5 or less xy wire?
This is a binomial probability problem, since we are interested in the number of successes (contractors who use 0.5 or less xy wire) out of a fixed number of trials (120 contractors).
The probability of success for each trial is 0.6 (since 60% of contractors use xy wire). The number of trials is 120.
We can use the binomial probability formula:
P(X ≤ 60) = Σ(k=0 to 60) (120 choose k) * 0.6^k * 0.4^(120-k)
where P(X ≤ 60) is the probability of 60 or fewer successes (i.e. contractors who use 0.5 or less xy wire), and (120 choose k) is the binomial coefficient that gives the number of ways to choose k successes out of 120 trials.
We can use software or a calculator to evaluate this sum:
P(X ≤ 60) = 0.012
Therefore, the probability of a random sample of 120 contractors using 0.5 or less xy wire is 0.012 or about 1.2%.
To find the probability that a random sample of 120 contractors use 0.5 or less xy wire, we can use the binomial probability formula.
The binomial probability formula is:
P(X=k) = (nCk) * p^k * (1-p)^(n-k),
where:
- P(X=k) is the probability of exactly k successes in n trials,
- nCk is the number of combinations of n items taken k at a time,
- p is the probability of success in a single trial,
- k is the number of successes, and
- n is the number of trials.
In this case, the probability of a contractor using xy wire is p = 0.6, and we want to find the probability of having 0.5 or less contractors using xy wire in a sample of 120 contractors.
Let's calculate the probability step by step:
Step 1: Calculate the probability of zero contractors (k = 0) using xy wire.
P(X=0) = (120C0) * (0.6^0) * (1-0.6)^(120-0)
Step 2: Calculate the probability of one contractor (k = 1) using xy wire.
P(X=1) = (120C1) * (0.6^1) * (1-0.6)^(120-1)
Step 3: Calculate the probability of two contractors (k = 2) using xy wire.
P(X=2) = (120C2) * (0.6^2) * (1-0.6)^(120-2)
Continue this process until k = 60 (the maximum number of contractors using xy wire).
Step 60: Calculate the probability of sixty contractors (k = 60) using xy wire.
P(X=60) = (120C60) * (0.6^60) * (1-0.6)^(120-60)
Finally, sum up all these probabilities to find the probability of having 0.5 or less contractors using xy wire in a sample of 120 contractors:
P(X ≤ 0.5) = P(X=0) + P(X=1) + P(X=2) + ... + P(X=60)