An electronics company is about to launch a new product. If the serial number for each piece produced has the following format: LLNNN where L stands for any letter in the English alphabet and N is a number from 0 to 9, please answer the following:
a)What is the name of the counting rule used to find all the possible outcomes?
b)How many different items can be uniquely numbered?
2)
(5 pts)
A company needs to select an expert group of consultants to provide advice for a given project. How many possible selections consisting of 2 project managers, 2
legal advisors, 4 computer scientists and 3 systemsengineers can be selected if the company can pick from 4 project managers, 7 legal advisors, 5 computer scientists and 6 systems engineers? Think carefully about the counting rules involved before attempting to do any calculations.
3)
(5 pts)
Consider that you are a line manager in your current Corporation. There is a 0.50 probability that you will be promoted this year. There is a 0.65 probability that you will get a promotion or a raise. The probability of getting a promotion and a raise is 0.35. (Use letter P to denote promotion and letter R to denote raise.)
a. If you get a promotion, what is the probabilitythat you will also get a raise?
b. What is the probability that you will get a raise?
c. Are getting a raise and being promoted independent events? Explain using
probabilities.
d. Are these two events mutually exclusive? Explain using probabilities.
Please show all the steps in your calculations
Please display answer here
1)
a) The counting rule used to find all the possible outcomes is the multiplication principle or rule of product. According to this rule, the total number of outcomes is obtained by multiplying the number of choices for each individual element of an event. In this case, there are 26 choices for the first letter (any letter in the English alphabet), 26 choices for the second letter (any letter in the English alphabet), and 10 choices for each of the three digits. Therefore, the total number of possible outcomes is calculated as 26 * 26 * 10 * 10 * 10 = 676,000.
b) Since there are 26 choices for each of the two letters and 10 choices for each of the three digits, the total number of different items that can be uniquely numbered is calculated as 26 * 26 * 10 * 10 * 10 = 676,000.
2)
To find the number of possible selections, we need to use the combination rule, which calculates the number of ways to select items from a larger set without regard to order.
The number of different selections of 2 project managers from 4 is calculated as C(4, 2) = 4! / (2! * (4-2)!) = 6.
The number of different selections of 2 legal advisors from 7 is calculated as C(7, 2) = 7! / (2! * (7-2)!) = 21.
The number of different selections of 4 computer scientists from 5 is calculated as C(5, 4) = 5! / (4! * (5-4)!) = 5.
The number of different selections of 3 systems engineers from 6 is calculated as C(6, 3) = 6! / (3! * (6-3)!) = 20.
Now, we need to multiply the calculated values for each category together to find the total number of possible selections:
Total number of possible selections = number of project manager selections * number of legal advisor selections * number of computer scientist selections * number of systems engineer selections
= 6 * 21 * 5 * 20 = 12,600.
Therefore, there are 12,600 possible selections of consultants for the given project.
3)
a) The probability of getting a promotion and a raise is denoted as P(P and R) = 0.35.
The probability of getting a promotion is denoted as P(P) = 0.50.
To find the probability of getting a raise given that you get a promotion, we use the formula:
P(R|P) = P(P and R) / P(P) = 0.35 / 0.50 = 0.70 or 70%.
Therefore, the probability of getting a raise given that you get a promotion is 0.70 or 70%.
b) The probability of getting a promotion or a raise is denoted as P(P or R) = 0.65.
To find the probability of getting a raise, we need to subtract the probability of not getting a raise from the probability of getting a promotion or a raise:
P(R) = P(P or R) - P(P and not R) = 0.65 - P(P and not R).
To find P(P and not R), we subtract the probability of getting a promotion and a raise from the probability of getting a promotion:
P(P and not R) = P(P) - P(P and R) = 0.50 - 0.35 = 0.15.
Now we can calculate P(R):
P(R) = 0.65 - 0.15 = 0.50 or 50%.
Therefore, the probability of getting a raise is 0.50 or 50%.
c) Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other event. In this case, we need to compare the probability of getting a raise (R) with the probability of being promoted (P) to determine if they are independent.
If two events are independent, then P(R) = P(R|P).
But in this case, P(R) = 0.50 and P(R|P) = 0.70.
Since P(R) is not equal to P(R|P), we can conclude that getting a raise and being promoted are not independent events.
d) Two events are mutually exclusive if they cannot occur at the same time. In this case, we need to determine if getting a raise (R) and being promoted (P) are mutually exclusive by comparing their probabilities.
If two events are mutually exclusive, then P(R and P) = 0.
But in this case, P(R and P) = 0.35.
Since P(R and P) is not equal to 0, we can conclude that getting a raise and being promoted are not mutually exclusive events.