1. Main concept. Definition of probability

Concept about random events

When we are calculating probability, we are concerned with the chance of one particular event from that sample space occurring. Events can be simple or compound. A simple event results in just one outcome. For instance, if we flip one coin, it will result in just one outcome. The coin could either land on heads, or it could land on tails. A compound event is an event containing more than one outcome. For example: rolling a 2 followed by rolling an odd number with a die. Events can sometimes be labeled as complementary events. One event is complementary to another when that event can occur if, and only if, the other does not occur. Again, let’s consider a coin toss with just one coin. The complement of flipping heads would be flipping tails. One can only occur if the other does not occur. Another way of saying this is that events are complementary if their sample space includes all possible outcomes, meaning that the sum of their probabilities is 1. Compatible events are events that can occur at the same time. Incompatible events cannot occur at the same time. If events A and B are incompatible, it is impossible for both A and B to happen at the same time. For instance, if you are rolling a typical die and event A is to roll a 1, 2, or 3 and event B is to roll a 4, 5, or 6, then events A and B are incompatible. It is not possible for both A and B to occur at the same time. If, however, event A is to roll an even number and event B is to roll a number larger than 3, then events A and B are compatible, because it is possible for both events to occur at the same time. Finally, probability events can be grouped as either dependent or independent. With dependent events, the probability of the second outcome can change based on the results of the first. Two events are said to be independent when the outcome of one event does not change the probability of the other. If you are drawing two cards from a regular deck of playing cards, drawing one card out at a time and replacing that card back into the deck each time, then the outcome of the first draw would not affect the outcome of the second. These would be independent events. If, however, you were not replacing the card that you drew out on the first draw, then the probability of the second draw would be affected by the first event. There would be fewer cards remaining for the second draw, and the card that you drew on your first turn would not be available the second time. These would then be dependent events. Denise is conducting a probability experiment, using the two spinners shown here. She is going to spin both spinners at the same time and wants to determine the probability of the first spinner landing on white (Event A) and the probability of the second spinner landing on grey (Event B).

Four Basic Types of Events

• Mutually Exclusive Events: These are events that cannot occur at the same time. The cause of mutually exclusive events could be a force of nature or a man made law. Being twenty-five years old and also becoming president of the United States are mutually exclusive events because by law these two events cannot occur at the same time.

• Complementary Events: These are events that have two possible outcomes. The probability of event A plus the probability of A' equals one. P(A) + P(A') = 1. Any event A and its complementary event A' are mutually exclusive. Heads or tails in one toss of a coin are complementary events.

• Independent Events: These are two or more events for which the outcome of one does not affect the other. They are events that are not dependent on what occurred previously. Each toss of a fair coin is an independent event.

• Conditional Events: These are events that are dependent on what occurred previously. If five cards are drawn from a deck of fifty-two cards, the likelihood of the fifth card being an ace is dependent on the outcome of the first four cards.

Classical definition of probability

Probability is defined as the chance that an event will happen or the likelihood that an event will happen.

The definition of probability is

.

The favorable events are the events of interest. They are the events that the question is addressing. The total events are all possible events that can occur relevant to the question asked. In this definition, favorable has nothing to do with something being defective or non-defective.

What is the probability of a head occurring in one toss of a coin?

The number of favorable events is 1 (one head) and the number of total events is 2 (head or tail). In this case, the probability formula verifies what is obvious.

Probability numbers always range from 0 to 1 in decimals or from 0 to 100 in percentages.

Notation for Probability Questions

Instead of writing out the whole question, the following notation is used.

• What is the probability of event A occurring? = Probability (A) = P(A)

• What is the probability of events A and B occurring? = P(A and B) = P(A) and P(B)

• What is the probability of events A or B occurring? = P(A or B) = P(A) or P(B)

Probability in Terms of Areas

Probability may also be defined in terms of areas rather than the number of events.

.

Methods to determine probability values

There are three major methods used to determine probability values.

• Subjective Probability: This is a probability value based on the best available knowledge or maybe an educated guess. Examples are betting on horse races, selecting stocks or making product-marketing decisions.

• Priori Probability: This is a probability value that can be determined prior to any experimentation or trial. For example, the probability of obtaining a tail in tossing a coin once is fifty percent. The coin is not actually tossed to determine this probability. It is simply observed that there are two faces to the coin, one of which is tails and that heads and tails are equally likely.

• Empirical Probability: This is a probability value that is determined by experimentation. An example of this is a manufacturing process where after checking one hundred parts, five are found defective. If the sample of one hundred parts was representative of the total population, then the probability of finding a defective part is .05 (5/100). The question may be asked: How is it known that this sample is representative of the total population? If repeated trials average .05 defective, with little variation between trials, then it can be said that the empirical probability of a defective part is .05.

Main formulas of combinatorial calculus

Very often the number of outcomes for an event or the sample space is very large. In order to handle such large numbers it is useful to take advantage of certain counting techniques which also provide compact formulas.

Product rule.

Suppose an experiment consists of parts, where the first part has possible outcomes, the second , and so on. Then there are possible outcomes for the experiment.

Example 1. A dice is thrown twice or two dice are thrown at once. There are outcomes and the probability to get, e.g. the pair (2,5), is

Permutations.

Given different objects , we may ask for the number of possible different (ordered) arrangements, also called permutations, of these objects.

Example 2. For three objects a, b, c we have 6 possible arrangements abc, acb, bac, bca, cab, cba.

In the general case this number can be found as follows: For the first object we have n possibilities - once we have selected the first object, there remain possibilities for the second object. Proceeding in this way until the last object, we find that the number of possible permutations in this case is

.

The number is called n factorial (by convention one puts 0!=1).

By the same argument one finds that the number of different arrangements of objects selected from objects equals

.

Note that we have This number can also be expressed as

.

Example 3. An urn contains nine balls which are labeled with the numbers 1, …,9.

4 balls are drawn at random without replacement.

a) What is the probability to get the the numbers “2467’’ in this order.

b) What is the probability that the drawn numbers contain the numbers “345’’.

Solution:

a) Our event “2467’’ consists of one element and the sample space has elements, therefore we get

b) This event can be realized in 12 outcomes, namely “x345’’ or “345x’’, where x is a number from 1, …,9 different from 345. We thus get

Example 4. In a room there are 23 persons. What is the probability that aren’t two of them with the same birthday.

- Since the persons may have been born on any day of the year, we have

possible combinations of birthdays.

- In order that all birthdays be different, we have 365 choices for the first person,

364 for the second and so on. The searched for number therefore equals

356 and the probability is

In how many ways we can choose elements from different elements, if the order is not considered. In order to solve this problem, we can take the number of ordered arrangements of elements from elements and divide by the number of permutations of every group of elements which equals We thus obtain that the number of possible combinations of elements out of elements is given by the binomial coefficient

By convention one puts In particular, the binomial coefficient fulfills the relation

.

Example 5. Find the combinations of 3 elements from a, b, c, d. We have the following choices:

{a, b, c}; {a, b, d}; {a, c, d}; {b, c, d}.

Example. 6. cards are chosen from a pile of 52 cards. What is the probability to

get 4 aces.

- The number of possible outcomes equals the number of possible combinations of drawing 4 cards from a pile of 52 without taking care of the order, which is

the probability is therefore