**Probability sample questions, Worksheet, pdf, **

Counting principles in probability are the basic rule for counting the number of the elements in a set. It usually uses for counting the size of an event in probability and statistics. There are two major counting principles, namely **addition** and **multiplication**. based on these two fundamental principles some other counting methods can be introduced like permutation and combination. In this brief and comprehensive post, we explain how to use this counting principles.

*Addition Principle:*

The addition principles is used when two event cannot happen at same time. For example, we have to chose between to actions of flipping a coin or tossing a die (only one of these two actions not both of them) and the number of the possible results can be counted by the addition principle. In this example, the two actions of flipping the coin and tossing the die cannot happen simultaneously so the outcome will be the set {face, tail, 1, 2, 3, 4, 5, 6} which has

8= 2 (number of possible results of coin) + 6 (number of possible results of die).

The tree diagram for this experiment is

The numbers 1 to 6 in this diagram are the results which appear on the die and F and T letters are the results which appear on the coin. Since these two actions cannot happen at the same time the results will be added.

*Multiplication Principle:*

The addition principles is used when two event happen at same time. For example, we are going to flip a coin and toss a die (when it does not mention it means we do these two actions at the same time) and the number of the possible results can be counted by the multiplication principle. In this example, the two actions of flipping the coin and tossing the die happen simultaneously, so the outcome will be the set

** { ((F,1), (F,2), (F,3), (F,4), (F,5), (F,6), (T,1), (T,2), (T,2), (T,3), (T,4), (T,5), (T,6) }**

which has

12= 2 (number of possible results of coin) * 6 (number of possible results of die).

The tree diagram for this experiment is

*Example: *

In a drawer, there are 4 shirts, 5 pants and 8 pairs of socks. In how many ways a set of a cloth including a shirt, a pants and a pair of socks can be chosen?

Answer:

For selecting a set of cloth, all the three types of cloths must be selected in a same time. Therefore we can count the number of the possible ways by multiplication principle.

n = 4 (number of shirts) * 5 (number of pants) * 4 (number of pairs of socks) = 80

*Example: *

Three cities A, B and C are connected as the below.

The cities A and B are connected by the paths 1, 2 and 3 while the cities B and C are connected by paths 4, 5, 6 and 7. In how many ways a passenger can travel from city A to city B?

Answer:

For traveling from city A to city B, the passenger must chose one of the 3 paths between A and B and also chose one of the paths between B and C. In order to have a complete travel, this two task must be done simultaneously. Therefore the number of paths is calculated by multiplication principle.

Number of paths between A and C = 3 (number of paths between A and B) * 4 (number of paths between B and C) = 12.

These 12 paths are (1,4), (1,5), (1,6), (1, 7), (2,4), (2,5), (2,6), (2, 7) where the first number in each order pair represents the path between cities A and B and the second number represents the path between cities B and C.

**The pdf file below is a worksheet of 23 sample questions in this topics with solutions. **

## Comments