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Probability

Probability of an event

Basics of Probability: Till now, we learned about exploratory analysis of data, collection of data and now we begin the transition to inference. In order to do inference, we need to use the language of probability. In order to use the language of probability, we need an understanding of random variables. In this chapter, we learn about the basic rules of probability, what it means for events to be independent, discrete and continuous random variables, simulation and rules for combining random variables.

Probability is a measure of the likeliness that an event will occur. It is used to quantify an attitude of mind towards some proposition of whose truth we are not certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty), we call probability. Thus the higher the probability of an event, the more certain we are that the event will occur. A simple example would be the toss of a fair coin. Since the 2 outcomes are deemed equiprobable, the probability of "heads" equals the probability of "tails" and each probability is 1/2 or equivalently a 50% chance of either "heads" or "tails".

Experiment (Random phenomenon): An experiment [i.e., random phenomenon] is an activity in which we know what outcomes could happen, but we don't know which particular outcome will happen.

  • Example: If we toss a coin, we know that we will get head (or) tail, but we don't know which particular outcome will happen. Similarly, if we roll a six sided die, we know that we will get a 1, 2, 3, 4, 5 or 6, but we don't know which particular outcome will happen.

Trial and Outcome: A single attempt (or) realization of an experiment is known as trial and one of the possible results of an experiment is known as an outcome. For example: the possible outcomes for tossing a single coin is: head (or) tail; the possible outcomes for rolling a six-sided die is: 1, 2, 3, 4, 5 (or) 6.

Sample space: The set of all possible outcomes of an experiment is known as sample space. It is denoted by the capital letter 'S'. For example: sample space for the roll of a single die, S = {1, 2, 3, 4, 5, 6}; sample space for tossing a single coin, S = {H, T}.

  • Example: Find the sample space when two six sided dice are rolled simultaneously.

    Solution: When two dice are rolled simultaneously, there are total 36 possible outcomes. Therefore, sample space (S) = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}.

Event: An event is a collection of outcomes of an experiment, that is, an event is the subset of the sample space. For example: sample space for the roll of a single die, S = {1, 2, 3, 4, 5, 6}. Let event P = "face value of the die is an even number". Then P = {2, 4, 6}. Let event Q = "face value of the die is an odd number". Then Q = {1, 3, 5}. Events P and Q are subsets of the sample space.

Occurrence of events: In a random experiment, if E is the event of a sample space S and w is the outcome, then we say that the event E has occurred if w E. If w E, then we say that the event E has not occurred.

Important Types of Events

Simple event: Any event which consists of only a single outcome in the sample space is called simple event. It can also be known as elementary event. For example: in tossing a single coin experiment, obtaining head (or tail) is an elementary event because it consists of only a single outcome in the sample space, that is, {H} or {T}.

Compound event: Any event which consists of more than one outcome in the sample space is called compound event. It can also be known as mixed event. For example: in rolling a six-sided die experiment, obtaining an even number is a compound event because it consists of three outcomes in the sample space, that is, {2, 4, 6}.

Sure (or Certain) event: In a random experiment, an event will be called a sure event or a certain event if it always occurs whenever an experiment is performed. In other words, let S be a sample space and if E S, then E is an event called sure event.

For example, if we roll a six sided die, the possible outcomes of the sample space will be 1, 2, 3, 4, 5, 6. Now we consider an event of "Getting an even or odd number" in any particular event. We find that this event is represented with the following elements in the set 1, 2, 3, 4, 5, 6 which is exactly same as the sample space. So it is called sure event.

Equally likely events: Events are said to be equally likely, if we have no reason to believe that one is more likely to occur than the other. For example: when a six sided die is thrown, all the six-faces {1, 2, 3, 4, 5, 6} are equally likely to come-up.

Impossible event: In a random experiment, let S be a sample space and if S, then is an event called an impossible event. It is also known as null event. For example: in rolling a six-sided die experiment, obtaining the face value of the die above 6 or below 1 is an impossible event because sample space = {1, 2, 3, 4,5, 6}.

Mutually exclusive (or Disjoint) events: Two or more events are said to be mutually exclusive events if and only if they have no outcomes in common. For example: when a coin is tossed, the event of occurrence of a head and the event of occurrence of a tail are mutually exclusive events.

Complementary event: In a random experiment, let S be the sample space and E be an event. The complement of an event E with respect to S is the set of all the elements of S which are not in E. The complement of E is denoted by E' or Ec.

Probability

Probability of an event: It is defined as the relative frequency of an outcome. That is, it is the fraction of time that the outcome would occur if the experiment were repeated indefinitely. For any event A, probability is defined as: , where s = number of ways an outcome can succeed, f = number of ways an outcome can fail and (s + f) is the total number of outcomes in the sample space.

The probability of any event 'A' ranges from 0 to 1, inclusive. That is, 0 ≤ P(A) ≤ 1. This is an algebraic result from the definition of probability when success is guaranteed (f = 0, s = 1) or failure is guaranteed (f = 1, s = 0). The sum of the probabilities of all possible outcomes in a sample space is equal to one. That is, if the sample space is composed of 'n' possible outcomes, then .

  • Example: There are 4 white marbles and 4 blue marbles in a bag. If a marble is drawn from a bag, what is the probability that it is a white color marble?
  • Solution: In the experiment of drawing a marble, let the event E = obtain a white color marble. The sample space contains 8 marbles. Here, s = 4 because there are 4 ways for our outcome to be considered a success and f = 4. Therefore, required probability is:
    P(E) = .

Probabilities of combined events:

  • P(A or B): The probability that either an event A (or) an event B occurs. Using set notation, this can be written as P(A B). A B is spoken as: A union B.
  • P(A and B): The probability that both an event A and an event B occur. Using set notation, this can be written as P(A B). A B is spoken as: A intersection B.
  • Complement of an event A: Set of all the outcomes in the sample space that are not included in the outcomes of event A is known as complement of an event A. The complement of an event A can be represented by either Ac (or) A and it is read as "A complement (or) not A". The probability of complement of an event A is equal to the '1' minus the probability of an event A P(A) = 1 – P(A).

Probability of mutually exclusive events: Two or more events are said to be mutually exclusive events if and only if they have no outcomes in common. If A and B are two mutually exclusive events, then A B = Φ P(A B) = 0.

  • Example: In the two-dice rolling experiment, A = "face shows a 2" and B = "sum of the two dice is 9" are mutually exclusive because there is no way to get a sum of 9 if one die shows a 2. That is, events A and B cannot both occur.

Conditional probability: The probability of an event occurring given that another event has already occurred is known as conditional probability.

  • P(A given B): The probability of an event A occurring given that an event B has already occurred is known as conditional probability of 'A given B' and it is represented by . The formula for calculating the P(A given B) is: .
  • P(B given A): The probability of an event B occurring given that an event A has already occurred is known as conditional probability of 'B given A' and it is represented by . The formula for calculating the P(B given A) is: .

Some conditional probability problems can be solved by using a tree diagram. A tree diagram is a schematic way of looking at all possible outcomes.

Addition rule: According to this rule, probabilities of all the related events are added together to compute the probability of their joint occurrence. If A and B are any two events associated with a random experiment, then addition rule for A and B is given as: P(A or B) = P(A) + P(B) – P(A and B).

Special case of the addition rule: If A and B are two mutually exclusive events, then addition rule for A and B is given as: P(A or B) = P(A) + P(B).

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