In the past few years, outsourcing overseas has become more frequently used than ever before by 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.
a. What is the probability that exactly five companies that outsource overseas use a consultant?
b. What is the probability that more than eleven companies that outsource overseas use a consultant?
c. What is the probability that none of the companies that outsource overseas use a consultant?
d. What is the probability that between three and seven (inclusive) companies that outsource overseas use a consultant?
e. Construct a graph for this binomial distribution.
Prob(using consultant) = 20/100 = 1/5
prob(NOT using consultant) = 4/5
a) prob(exactly 5 of 15 use con) = C(15,5) (1/5)^5 (4/5)^10 = ...
b) find the probs for exactly 12, exactly 13 ... exactly 15 and add up those 4 cases.
c) find prob(that all 15 use a con) and subtract that from 1
d) What do you think?
e) all yours,
ok
To solve these problems, we can use the binomial probability formula:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of x successes
n is the number of trials (in this case, the number of companies selected)
x is the number of successes (the number of companies using a consultant)
p is the probability of success (the proportion of companies that use a consultant)
a. To find the probability that exactly five companies that outsource overseas use a consultant:
Here, n = 15 (total number of selected companies), x = 5 (number of companies using a consultant), p = 0.2 (proportion of companies that use a consultant)
P(5) = (15C5) * 0.2^5 * (1-0.2)^(15-5)
To calculate (15C5), we use the combination formula: (15C5) = 15! / (5! * (15-5)!)
P(5) = (15! / (5! * (15-5)!) * 0.2^5 * 0.8^10
Simplifying the calculation, we find:
P(5) ≈ 0.257
Therefore, the probability that exactly five companies that outsource overseas use a consultant is approximately 0.257.
b. To find the probability that more than eleven companies that outsource overseas use a consultant:
We need to calculate the probabilities of having exactly 12, 13, 14, and 15 companies using a consultant. Then, we sum these probabilities.
P(x > 11) = P(12) + P(13) + P(14) + P(15)
Using the same formula and values as in part (a), we can calculate each probability individually and sum them up.
P(x > 11) ≈ P(12) + P(13) + P(14) + P(15) ≈ 0.003 + 0.0004 + 0.0001 + 0.00001
Therefore, the probability that more than eleven companies that outsource overseas use a consultant is approximately 0.00341.
c. To find the probability that none of the companies that outsource overseas use a consultant:
Here, x = 0 (number of companies using a consultant), p = 0.2 (proportion of companies that use a consultant)
P(0) = (15C0) * 0.2^0 * 0.8^(15-0)
P(0) = (15C0) * 0.2^0 * 0.8^15
Using the combination formula again:
P(0) = (15! / (0! * (15-0)!) * 0.2^0 * 0.8^15
P(0) ≈ 0.035
Therefore, the probability that none of the companies that outsource overseas use a consultant is approximately 0.035.
d. To find the probability that between three and seven (inclusive) companies that outsource overseas use a consultant:
We need to calculate the probabilities of having exactly 3, 4, 5, 6, and 7 companies using a consultant. Then, we sum these probabilities.
P(3 ≤ x ≤ 7) = P(3) + P(4) + P(5) + P(6) + P(7)
Using the same formula and values as in part (a), we can calculate each probability individually and sum them up.
P(3 ≤ x ≤ 7) ≈ P(3) + P(4) + P(5) + P(6) + P(7) ≈ 0.136 + 0.228 + 0.257 + 0.212 + 0.114
Therefore, the probability that between three and seven (inclusive) companies that outsource overseas use a consultant is approximately 0.947.
e. The graph for the binomial distribution can be constructed by plotting the probability of each possible outcome (0 to 15 companies using a consultant) on the y-axis and the number of companies using a consultant on the x-axis. The graph will have a shape resembling a bell curve.
Note that the probabilities mentioned in parts (a), (b), (c), and (d) can be plotted on the graph to visualize the distribution.
To answer these questions, we need to use the binomial probability formula. The formula is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes (companies using a consultant),
- n is the number of trials (number of companies selected),
- k is the number of successes (companies using a consultant),
- (n C k) is the combination formula (n choose k),
- p is the probability of success in a single trial (probability of a company using a consultant), and
- (1 - p) is the probability of failure in a single trial (probability of a company not using a consultant).
a. What is the probability that exactly five companies that outsource overseas use a consultant?
To find the probability of exactly five companies using a consultant, we substitute the values into the binomial formula:
P(X = 5) = (15 C 5) * (0.2^5) * (0.8^(15 - 5))
Using the combination formula (15 C 5) = 3003, the probability of exactly five companies using a consultant is:
P(X = 5) = 3003 * (0.2^5) * (0.8^10)
b. What is the probability that more than eleven companies that outsource overseas use a consultant?
To find the probability of more than eleven companies using a consultant, we need to calculate the probabilities of 12, 13, 14, and 15 companies using a consultant and sum them up:
P(X > 11) = P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)
Using the same formula as in part a, calculate each probability and sum them.
c. What is the probability that none of the companies that outsource overseas use a consultant?
To find the probability of none of the companies using a consultant, we substitute the values into the binomial formula:
P(X = 0) = (15 C 0) * (0.2^0) * (0.8^15)
Using the combination formula (15 C 0) = 1, the probability of none of the companies using a consultant is:
P(X = 0) = 1 * (0.2^0) * (0.8^15)
d. What is the probability that between three and seven (inclusive) companies that outsource overseas use a consultant?
To find the probability of between three and seven companies using a consultant, we need to sum up the individual probabilities for 3, 4, 5, 6, and 7 companies using a consultant:
P(3 ≤ X ≤ 7) = P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)
Using the same formula as in part a, calculate each probability and sum them.
e. Construct a graph for this binomial distribution.
To construct a graph for the binomial distribution, you can plot the probabilities for various values of k (number of companies using a consultant) on the x-axis and the corresponding probabilities on the y-axis. The values of k will range from 0 to the maximum possible value (in this case, 15). Plot the individual probabilities for each value of k and connect them to create a histogram-like bar graph.