There are two fundamental principles of counting; Fundamental principle of addition and fundamental principle of multiplication. I often got confused applying them. I know that if there are two jobs, say m and n, such that they can be performed independently in m and n ways respectively, then either of the two jobs can be performed in m+n ways and when two jobs are performed in succession, they can be performed in m×n ways. My question is how to identify whether jobs are independent or in succession? Is there any simple way to identify this?

Graham Beasley 2022-07-19 Answered
There are two fundamental principles of counting; Fundamental principle of addition and fundamental principle of multiplication.

I often got confused applying them. I know that if there are two jobs, say m and n, such that they can be performed independently in m and n ways respectively, then either of the two jobs can be performed in m + n ways and when two jobs are performed in succession, they can be performed in m × n ways.

My question is how to identify whether jobs are independent or in succession?

Is there any simple way to identify this? Are there any keywords?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

bgr0v
Answered 2022-07-20 Author has 13 answers
The distinction is not between "independently" and "in succession". The real distinction is whether you are doing both, or you are doing just one.

The number of ways to do one of the jobs is m+n (either do the first job, which can be done in m ways; or do the second job, which can be done in n ways; total number of ways, m+n).

The number of ways of doing both is mn, because you have m ways of doing the first job, and n ways of doing the second job, and you have to do both.

Say you have 5 pants and 3 shirts. If you are giving one piece of clothing away, then you have 5+3 ways of deciding which piece to give away. But if you are deciding what to wear, you need to pick a pair of pants, and a shirt. That gives you 5×3 possible combinations. You can make the choices simultaneously (they don't have to be "in succession"). The "independence" clause is jut that choice of shirt/pants should not restrict your choice of pants/shirt (so you don't have a plaid shirt and a pair of striped pants that cannot be worn together...). What you choose for job one does not affect what you can choose for job two.

So you have to think about what you are doing. I'm not sure if there are "keywords" that you should be looking at.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-09-10
Sam and Allen have applied for scholarships. The probability that Sam gets a scholarship is 0.93. The probability that Allen gets a scholarship is 0.77. The probability they both get scholarships is 0.74.
Whats the probabilty that Sam gets a scholarship but Allen doesnt?
asked 2022-02-13
At a local university, 54.3% of incoming first year students have a PC. If 3 students are selected randomly, what are the probabilities that at least one has a computer?
asked 2022-02-14
What is the probability of getting three heads or three tails with three coin flips?
asked 2021-11-21
Suppose a die is weighted such that the probability of rolling a three is the same as rolling a six, the probability of rolling a one, two, or four is 3 times that of six, and the probability of rolling a five is 5 times that of rolling a three. Find the probability of 1. rolling a one 2. rolling a two 3. rolling a three 4. rolling a four 5. rolling a five 6. rolling a six
asked 2022-04-06
The same three digit prefix is used for all of the telephone numbers in a small town. How many different telephone numbers are possible by changing only the last four digits?
asked 2021-11-18
The monthly worldwide average number of airplane crashes of commercial airlines is 3.5. What is the probability that, there will be (a) at least 2 such accidents in the next month; (b) at most 1 accident in the next month? Explain your reasoning!
asked 2021-08-15
A binomial probability experiment is conducted with the given parameters.
Compute the probability of x successes in the n independent trials of the experiment.n=11,p=0.75,x8

New questions

The Porsche Club of America sponsors driver education events that provide high-performance driving instruction on actual racetracks. Because safety is a primary consideration at such events, many owners elect to install roll bars in their cars. Deegan Industries manufactures two types of roll bars for Porsches. Model DRB is bolted to the car using existing holes in the car's frame. Model DRW is a heavier roll bar that must be welded to the car's frame. Model DRB requires 20 pounds of a special high alloy steel, 40 minutes of manufacturing time, and 60 minutes of assembly time. Model DRW requires 25 pounds of the special high alloy steel, 100 minutes of manufacturing time, and 40 minutes of assembly time. Deegan's steel supplier indicated that at most 40,000 pounds of the high-alloy steel will be available next quarter. In addition, Deegan estimates that 2000 hours of manufacturing time and 1600 hours of assembly time will be available next quarter. The pro?t contributions are $200 per unit for model DRB and $280 per unit for model DRW. The linear programming model for this problem is as follows:
Max 200DRB + 280DRW
s.t.
20DRB + 25DRW 40,000 Steel Available
40DRB + 100DRW ? 120,000 Manufacturing minutes
60DRB + 40DRW ? 96,000 Assembly minutes
DRB, DRW ? 0
Optimal Objective Value = 424000.00000
Variable Value blackuced Cost
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
DRB 1000.00000 0.00000
DRW 800.00000 0.00000
Constraint Slack/ Surplus Dual Value
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 0.00000 8.80000
2 0.00000 0.60000
3 4000.00000 0.00000
Objective Allowable Allowable
Variable Coef?cient Increase Decrease
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
DRB 200.00000 24.00000 88.00000
DRW 280.00000 220.00000 30.00000
RHS Allowable Allowable
Constraint Value Increase Decrease
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 40000.00000 909.09091 10000.00000
2 120000.00000 40000.00000 5714.28571
3 96000.00000 Infnite 4000.00000
a. What are the optimal solution and the total profit contribution?
b. Another supplier offeblack to provide Deegan Industries with an additional 500 pounds of the steel alloy at $2 per pound. Should Deegan purchase the additional pounds of the steel alloy? Explain.
c. Deegan is considering using overtime to increase the available assembly time. What would you advise Deegan to do regarding this option? Explain.
d. Because of increased competition, Deegan is considering blackucing the price of model DRB such that the new contribution to profit is $175 per unit. How would this change in price affect the optimal solution? Explain.
e. If the available manufacturing time is increased by 500 hours, will the dual value for the manufacturing time constraint change? Explain.