Probability and Normal Distribution
A probability is a numerical value that measures the uncertainty that a particular event will occur. Probability is one of the most important conceptual tools to assess the degree of uncertainty and thereby to reduce risk.
The possibility of an event ordinarily represents the proportion of times under identical circumstances that the event can be expected to occur. Such a long-run frequency of occurrences referred to as objective probability. In tossing a fair coin, the probability is ½ for getting a head. This can be verified by tossing the coin many times-heads will appear about half of the time. However, a probability value is often subjective, set solely on the basis of personal judgement. Subjective probability is used for events having meaningful long-run frequency of occurrence.
There is a difference between probability and possibility/chance. Probability can be measured, whereas possibility/chance is not measurable. Probability 0 doesn’t always mean absolute impossibility.
Basic Outcomes and Sample Space:
The possible outcomes of a random experiment are called the basic outcomes, and the set of all basic outcomes is called the sample space. The symbol S will be used to denote the sample space.
An example: What is the sample space for a roll of a single six-sided die?
S = [1, 2, 3, 4, 5, 6]
Measurement of Probability:
There are three approaches to measuring the probability:
(1) Classical or Priori Approach
(2) Empirical or Relative Frequency Approach
(3) Subjective Approach
(1) Classical or Priori Approach:
to probability is the assignment of probability values on the basis of theoretical expectations in a hypothetical population.
Number of outcomes favoring event A
p(A) = ———————————————————
Total number of possible outcomes
For example, since there are six mutually exclusive and equally likely outcomes of tossing a single die, the probability of obtaining one specific outcome (for example, a three) can be measured in the following way.
p(3) = ———
Similarly, the probability of obtaining a head (H) on a single toss of a perfectly balanced coin is:
p (H) = ———-
(2) Empirical or Relative Frequency Approach:
to probability is the assignment of probability values based on the observed relative frequency with which that outcome occurred in the past.
Number of observed outcomes favoring A
p (A) = —————————————————————
Total number of observed outcomes
Thus, if 120,463 silicon chips are examined and 463 are found defective, then probability of defectiveness [ p(A) ] may be estimated as
P(A) = —————– = 0.0038
(3) Subjective Approach:
to probability is the assignment of probability values based on the individual’s belief or confidence that a particular outcome will occur. For example, the director of personnel must assess the degree of success associated with various job application procedures; or the structure of a real estate deal may reflect anticipated mortgage rates one year from now; or the production supervisor of an automobile manufacturer may weigh the introduction potential for a new series of energy-efficient cars.
Certain and Impossible Events
From our basic definition of probability, it may be noted that a probability will always be between 0 and 1, inclusively [0<= p (A) <=1]. This is because the numerator in the probability fraction can never be negative nor can it be larger than the denominator. Two important observations follow from the definition. First, an event that is certain to occur will have the same value in both the numerator and the denominator, for the same events will result every time (that is, with frequency 1). Thus
P [ certain event ] = 1
The event “the next President of the united states will be at least 35 years old” is a certain event and has a probability of 1, because the Constitution specifies a minimum age of 35. The event “food prices will rise, fall, or remain unchanged” Likewise is certain and has a probability of 1.
At the other extreme, an impossible event’s frequency ratio will always have a 0 in the numerator, for such event will never occur. Thus,
P [ impossible event ] = 0
For example, the U.S. auto industry has limited capacity to produce cars and cannot make 1000 of them in any one year. This outcome for next year is impossible and we can state that
P [ 1000 cars made ] = 0
Events and Rules:
Three types of events are there and two rules are there.
Addition Rule: The addition rule states that p(A or B) = p(A) + p(B) – p(AB). p(A or B) refers to the probability of A or B or both occurring. p(AB) refers to the joint probability of A and B occurring together. The subtraction of p(AB) to prevent the double counting of elements A and B where they overlap.
(1) Mutually Exclusive Events:
are two or more outcomes that cannot occur simultaneously. The figure below illustrates the nature of mutually exclusive events. An event may be either A or B but not both A and B. A coin when flipped will land either heads or tails but not both. Only one number can occur in a single spin of a roulette wheel, because the ball can fall into only one slot. These are examples of mutually exclusive events.
Our addition rule for the probability of A or B now becomes p (A or B) = p (A)+ p(B), as p(AB)=0 for mutually exclusive events.
Problem 1. An MBA applies for a job in two firms X and Y. The probability of his being selected in firm X is 0.7 and being rejected at Y is 0.5. The probability of at least one of his applications being rejected is 0.6. What is the probability that he will be selected in one of the firms?
Multiplication Rule: The multiplication rule is used to determine the probability of an outcome, which requires the successive or simultaneous occurrence of several events. We need to consider two subcases, one for dependent events and one for independent events.
2. Dependent Events:
The multiplication rule for dependent events states that
p(AB) = p(A)p(BA)
The new symbol, p(BA), is that of a conditional probability and is read “the probability of B occurring given that the event A has occurred.
3. Independent Events:
Two events are independent when the probability of one occurring is unaffected by the occurrence of the other event. The multiplication rule for independent events obviously becomes
p(AB) = p(A)p(B)
A quantitative variable that varies in a chance or random way. For example,
Should you buy house today or wait until interest rates go down? If you represent management, should you accept union demands or take a chance on loosing production because of a strike?
A theoretical distribution of probability values associated with a random variable.
Continuous random variable
A random variable in which there are an unlimited number of possible fractional values.
Discrete random variable
A random variable characterized by gaps in which no real values are found.
PROBABILITY DISTRIBUTIONS AND EXPECTED VALUE OF DISCRETE RANDOM VARIABLES
Let us imagine that you are the dispatcher of repair trucks for an automobile rental agency in a large metropolitan region. One of your problems is to anticipate the demand for repair services so that you have sufficient trucks and personnel available to meet the expected demand. You collect data on the number of requests per day over a 100-day period. Table 1 shows this historical data on demand for repair services of an auto rental agency over a 100-day period. By dividing each frequency by total frequency, we arrive at a record of the probability (relative frequency) of past demand on any given day.
Demand for repair services of an auto rental agency
A form of weighted average in which the probabilities associated with each value of the random variable are used as weights; long-term average over many samplings. Symbolized by E(x), the expected value of a variable is the mean of the probability distribution. This is, in turn, the mean of the variable. Thus
E(X)= Xp(X)=mean of X
The example shows that on the average, you should expect approximately seven (6.89) daily requests for repair.
A probability distribution is a special form of frequency distribution where the frequencies take the form of probabilities and the basic variable is a collectively exhaustive, mutually exclusive list of possible events or outcomes. Probability distributions may be expressed in the form of an equation or presented as a table.
Two discrete probability distributions are:
(1) Binomial Distribution: If a random or chance event may take either of two forms, say A or not-A (A) and p(A) = p and p(A) = q = 1 – p, on a single trial and we repeat the experiment for n independent trials, recording as x the number of A’s appearing in the n trials, the binomial distribution will give the probabilities that various numbers of A will appear.
(2) Poisson Distribution: If an event can occur repeatedly at random over a large area or long period of time, the Poisson distribution may be used to determine the probability of its occurrence in a small area or short interval of time.
is a useful continuous probability distribution. The normal curve is a theoretical mathematical curve. Normal curve is used for normal distribution. For practical purposes normal curve should be converted into a standard normal curve and a given variable needs to be converted into a standard normal variate.
A large public utility maintains a car pool for the use of its employees when travelling on company business. Company records indicate that the cost per mile of operating company’s cars is normally distributed around a mean of 17.9 cents, with a standard deviation of 0.9 cent.
(a) What portion of the company’s cars costs less than 16.5 cents to operate?
(b) What portion of the company’s cars costs between 17.5 and 18.5 cents to operate?
(c) What portion of the company’s cars costs between 18.0 and 19.0 cents to operate?
(d) The most expensive 5 percent will have an operating cost of at least how many cents?
(e) The middle 80 percent of the cars’ operating costs will lie between what two cost figures?
Analysis of past data has shown a manufacturing firm that the hub thickness of a particular type of gear it manufactures is normally distributed around a mean thickness of 10 cm, with a standard deviation of 2 cm.
i. In a production of 7500 such gears, how many will have a thickness between 10.275 and 12 cm?
ii. How many will have a thickness greater than 9 cm?
iii. The thickest 5000 gears will exceed what thickness?
iv. The thinnest 375 gears will be less than what thickness?
A survey revealed that the profit earned by 2500 companies was normally distributed with a mean profit of Tk. 20 Lacs and standard deviation of Tk. 5 Lacs. Find:
i. How many of the companies earned profit between Tk. 20 Lacs and Tk. 26 Lacs?
ii. The number of companies earning profit between Tk. 23 Lacs and 27 Lacs.
iii. The number of companies earning profit between Tk. 18 Lacs and 26.5 Lacs.
iv. The number of companies earning profit more than Tk. 17.5 Lacs.
v. What is the probability that a company earns between Tk. 16 Lacs and Tk. 17 Lacs as profit?
vi. What was the highest profit of the lowest 5% of the companies?
Analysis of data on loan disbursement of the BKB indicates that the average amount of loan received by a borrower was estimated at Tk. 50 Lac, with a standard deviation of Tk. 5 Lac. Find:
i. The % of borrowers receiving loan between Tk. 50 Lac and Tk. 63 Lac
ii. The % of borrowers receiving loan between Tk. 55 Lac and Tk. 62.5 Lac
iii. The % of borrowers receiving loan between Tk. 47 Lac and Tk. 64 Lac
iv. The % of borrowers receiving loan Tk. 57 Lac or more
v. What is the lowest amount of loan received by the highest 5% of the borrowers?
vi. What is the probability that a borrower received loan between Tk. 42 Lac and Tk. 49 Lac?