In the past few years, outsourcing overseas has become more frequently used than ever before by Canadian and U.S. companies. However, outsourcing is not without problems. A recent survey by Purchasing magazine indicates that 20% of the companies that outsource overseas use a consultant. Suppose 15 companies that outsource overseas are randomly selected.
What is the probability that exactly five companies that outsource overseas use a consultant?
this is a binary probability ... use / don't use
(d + u)^15 = d^15 + 15 d^14 u + ... + 15 d u^14 + u^15
you want the 6th term ... 15C5 d^10 u^5 = 3003 * .8^10 *.2^5
To find the probability that exactly five companies out of 15 that outsource overseas use a consultant, we can use the binomial probability formula. The formula for the probability of exactly "x" successes in "n" trials is:
P(X = x) = (nCx) * p^x * (1 - p)^(n - x)
Where:
- n is the total number of trials (15 in this case),
- x is the number of desired successes (5 in this case),
- nCx represents the number of combinations of n items taken x at a time,
- p is the probability of success in a single trial (20% or 0.2 in this case).
So, let's calculate the probability:
P(X = 5) = (15C5) * (0.2)^5 * (1 - 0.2)^(15 - 5)
Using the combination formula (nCx = n! / (x! * (n - x)!)):
P(X = 5) = (15! / (5! * (15 - 5)!)) * (0.2)^5 * (0.8)^10
Calculating the combination:
P(X = 5) = (3003) * (0.2)^5 * (0.8)^10
Calculating the exponents:
P(X = 5) = 3003 * 0.00032 * 0.10737
Calculating the multiplication:
P(X = 5) = 0.102218624
Therefore, the probability that exactly five out of fifteen companies that outsource overseas use a consultant is approximately 0.102 or 10.2%.
To find the probability that exactly five companies out of the 15 selected use a consultant, we can use the binomial probability formula.
The formula for binomial probability is:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of getting exactly k successes
n is the number of trials (companies selected)
k is the number of desired successes (companies using a consultant)
p is the probability of success (percentage of companies that outsource overseas and use a consultant)
(1-p) is the probability of failure (percentage of companies that outsource overseas and do not use a consultant)
(nCk) is the combination formula, which calculates the number of ways to choose k successes from n trials (n choose k).
In this case, n = 15, k = 5, p = 0.20, and (1-p) = 0.80.
Using these values, we can calculate the probability as follows:
P(X = 5) = (15C5) * (0.20)^5 * (0.80)^(15-5)
Now, let's calculate each component of this formula step by step.
(15C5) = 15! / (5! * (15-5)!) = (15 * 14 * 13 * 12 * 11) / (5 * 4 * 3 * 2 * 1) = 3,003
(0.20)^5 = 0.00032
(0.80)^(15-5) = 0.8^10 = 0.1073741824
Now, we can substitute these values back into the formula:
P(X = 5) = 3,003 * 0.00032 * 0.1073741824
Calculating this expression gives us:
P(X = 5) ≈ 0.032
So, the probability that exactly five companies out of the 15 selected use a consultant is approximately 0.032, or 3.2%.