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.