Module 8.1 Probability Concept

Likelihood

A conditional probability of an occurrence is also called its likelihood.

The odds that an event will or will not occur is an alternative way of expressing probabilities. Consider an event that has a probability of occurrence of 0.125

0.125 / (1-0.125) =1/7 The odds for the event occurring are one-to-seven. The odds against the event occurring are the reciprocal of 1/7, which is seven-to-one. 1-0.125 / 0.125= 7

The probability that the DJIA will increase tomorrow is 2/3. The probability of an increase in the DJIA stated as odds is:

Odds for E= P(E)/ 1 - P(E)= 2/3 : 1/3= 2/1

Calculating a Joint Probability of any Number of Independent Events

On the roll of two dice, the joint probability of getting two 4s is calculated as: P(4 on first die AND 4 on second die) = P(4 on first die) × P(4 on second die) = 1/6 × 1/6 = 1/36 = 0.0278 On the flip of two coins, the probability of getting two heads is: P(heads on first coin and heads on second coin) = 1/2 × 1/2 = 1/4 = 0.25

Example: Multiplication rule of probability P(I) = 0.4, the probability of the monetary authority increasing interest rates (I) is 40%. P(R | I) = 0.7, the probability of a recession (R) given an increase in interest rates is 70%. What is P(RI), the joint probability of a recession AND an increase in interest rates?

P(RI)= P(R | I) x P(I) = 0.7 x 0.4= 0.28

If we know that the odds for an event are one-to-six, we can compute the probability of occurrence as

Probability of occurance: 1/ (1+6)= 0.1429= 14.29% Probability of NOT occur 6/(1+6)= 0.8571= 85.71%

The multiplication rule of probability

To determine the joint probability of two events: P(AB) = P(A | B) × P(B) That A will occur given B occurs (a conditional probability) and the probability that B will occur (the unconditional probability of B). This calculation is sometimes referred to as the multiplication rule of probability P(A | B)= P (AB)/ P(B)

The addition rule of probability

To determine the probability that at least one of two events will occur: P(A or B) = P(A) + P(B) − P(AB)

total probability rule

To determine the unconditional probability of an event, given conditional probabilities P(A) = P(A | B1)P(B1) + P(A | B2)P(B2) + ... + P(A | BN)P(BN) where B1, B2, ... BN is a mutually exclusive and exhaustive set of outcomes.

When dealing with independent events?

When dealing with independent events, the word and indicates multiplication, and the word or indicates addition. In probability notation:

A random variable, an outcome, an event, mutually exclusive events, and exhaustive events.

-A random variable is an uncertain quantity/number. -An outcome is an observed value of a random variable. -An event is a single outcome or a set of outcomes. -Mutually exclusive events are events that cannot both happen at the same time. -Exhaustive events are those that include all possible outcomes.

Empirical probability Priori probability Subjective probability

-An empirical probability is established by analyzing past data. -An a priori probability is determined using a formal reasoning and inspection process. -A subjective probability is the least formal method of developing probabilities and involves the use of personal judgment. An analyst may know many things about a firm's performance and have expectations about the overall market that are all used to arrive at a subjective probability, such as, "I believe there is a 70% probability that Acme Foods will outperform the market this year." Empirical and a priori probabilities, by contrast, are objective probabilities.

Two defining properties of probability.

-The probability of occurrence of any event (Ei) is between 0 and 1 (i.e., 0 ≤ P(Ei) ≤ 1). -If a set of events, E1, E2, ... En, is mutually exclusive and exhaustive, the probabilities of those events sum to 1 (i.e., ΣP(Ei) = 1).

Venn Diagram for Events That Are Not Mutually Exclusive

Illustrates the addition rule with a Venn Diagram and highlights why the joint probability must be subtracted from the sum of the unconditional probabilities. Note that if the events are mutually exclusive the sets do not intersect, P(AB) = 0, and the probability that one of the two events will occur is simply P(A) + P(B).

P(A | B) = P(A) × P(B) is only TRUE for

Independent event

Probability rolling any one of the numbers 1-6 with a fair die (each of the faces has the same probability of landing facing up)

Is 1/6 = 0.1667 = 16.7%. The set of events—rolling a number equal to 1, 2, 3, 4, 5, or 6—is exhaustive, and the individual events are mutually exclusive, so the probability of this set of events is equal to 1. We are certain that one of the values in this set of events will occur.

When dealing with independent events, the word and indicates multiplication, and the word or indicates addition. In probability notation:

P(A or B) = P(A) + P(B) − P(AB) the probability that either A or B will occur is simply the sum of the unconditional probabilities for each event, P(A or B) = P(A) + P(B). and P(A and B) = P(A) × P(B)

At a charity ball, 800 names are put into a hat. Four of the names are identical. On a random draw, what is the probability that one of these four names will be drawn?

P(name 1 or name 2 or name 3 or name 4) = 1/800 + 1/800 + 1/800 + 1/800 = 4/800 = 0.005

Example: Joint probability for more than two independent events What is the probability of rolling three 4s in one simultaneous toss of three dice?

Since the probability of rolling a 4 for each die is 1/6, the probability of rolling three 4s is: P(three 4s on the roll of three dice) = 1/6 × 1/6 × 1/6 = 1/216 = 0.00463

P(Ei)

The "probability of event i." If P(Ei) = 0, the event will never happen. If P(Ei) = 1, the event is certain to occur, and the outcome is not random.

Consider rolling a 6-sided die.

The number that comes up is a random variable. If you roll a 4, that is an outcome. Rolling a 4 is an event, and rolling an even number is an event. Rolling a 4 and rolling a 6 are mutually exclusive events. Rolling an even number and rolling an odd number is a set of mutually exclusive and exhaustive events.

conditional probability

Where the occurrence of one event affects the probability of the occurrence of another event. We might be concerned with the probability of a recession given that the monetary authority increases interest rates. the probability of A given the occurrence of B" is expressed as P(A | B)

Unconditional probability a.k.a. marginal probability

the probability of an event regardless of the past or future occurrence of other events. If we are concerned with the probability of an economic recession, regardless of the occurrence of changes in interest rates or inflation, we are concerned with the unconditional probability of a recession.