Three Approaches to Probability

Three Approaches to Probability 1. Classical Approach If an experiment has n simple outcomes, this method would assign a probability of 1/n to each outcome. In other words, each outcome is assumed to have an equal probability of occurrence. This method is also called the axiomatic approach. Example 1: Roll of a Die S = {1, 2, · · · , 6} Probabilities: Each simple event has a 1/6 chance of occurring. Example 2: Two Rolls of a Die S = {(1, 1), (1, 2), · · · , (6, 6)} Assumption: The two rolls are “independent.” Probabilities: Each simple event has a (1/6) · (1/6) = 1/36 chance of occurring. 2. Relative-Frequency Approach Probabilities are assigned on the basis of experimentation or historical data. Formally, Let A be an event of interest, and assume that you have performed the same experiment n times so that n is the number of times A could have occurred. Further, let nA be the number of times that A did occur.Now, consider the relative frequency nA/n. Then, in this method, we “attempt” to define P(A) as:

The above can only be viewed as an attempt because it is not physically feasible to repeat an experiment an infinite number of times. Another important issue with this definition is that two sets of n experiments will typically result in two different ratios. However, we expect the discrepancy to converge to 0 for large n. Hence, for large n, the ratio nA/n may be taken as a reasonable approximation for P(A).

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Example 1: Roll of a Die S = {1, 2, · · · , 6} Probabilities: Roll the given die 100 times (say) and sup- pose the number of times the outcome 1 is observed is 15. Thus, A = {1}, nA = 15, and n = 100. Therefore, we say that P(A) is approximately equal to 15/100 = 0.15. 3. Subjective Approach In the subjective approach, we define probability as the degree of belief that we hold in the occurrence of an event. Thus, judgment is used as the basis for assigning probabilities. Notice that the classical approach of assigning equal probabilities to simple events is, in fact, also based on judgment. What is somewhat different here is that the use of the subjective approach is usually limited to experiments that are unrepeatable. Rule of Addition 

The Addition Law says that given 2 events A, B

“At least one occurs”

P( A or B)  P( A)  P(B)  P( A and B) P( A  B)  P( A)  P(B)  P( A  B) “Neither of the events” P( A¹  B¹)  P( A  B)¹  1 P( A  B) “One occurs but not the other”

P( A B¹)  P( A)  P( A B)

Or rewrite as

P( A)  P( A B)  P( A B¹)

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Probability that event A does not occur P( A)  1  P( A) 

Independent events: Two events are independent if the occurrence of one of the events gives us no information about whether or not the other event will occur; that is, the events have no influence on each other.  P( A B)  P( A)  P(B)



Mutually Exclusive: Two events are mutually exclusive if it is impossible for them to occur together. P( A  B)  0

The Probability Distribution 1. Binomial Distribution can be used under the following conditions:    

The number of trials is fixed The trials are independent of each other There are two outcomes – success and failure The probability of success in each trial is constant.

2. Conditions for a Poisson distribution If X is the number of occurrences of a particular event in an interval of the fixed length in space or time, then the events occur:   

Independently of each other Singly in continuous space or time At a constant rate in the sense that the mean number in an interval is proportional to the length of the interval.

3. Poisson as an approximation to the binomial If X  

B(n, p) and n is large p is small

then X can be approximated by Po (np) . 4. Normal Approximations  Binomial Approximation

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The normal distribution can be used as an approximation to the binomial distribution, under certain circumstances, namely: 

If X ~ B(n, p) and if n is large and p is small, such that np > 5 and n(1-p) > 5 , then X is approximately N(np, npq) (where q = 1 - p).

In some cases, working out a problem using the Normal distribution may be easier than using a Binomial.  Poisson Approximation The normal distribution can also be used to approximate the Poisson distribution for large values of such that (the mean of the Poisson distribution). 

5. The Normal Distribution

Properties of a Normal Distribution     

The distribution is symmetrical about the mean  The mode, median and mean are all equal, due to the symmetry of the distribution. The range of X is from -  to +  The horizontal axis is asymptotic to the curve as x   and x  . The total area under the curve is unity.

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For a symmetric data,   

   includes almost 68 % of the observation i.e. P (    x     )  0.683   2 includes almost 95 % of the observation i.e. P (  2  x    2 )  0.95   3 includes almost all the observation i.e. P (  3  x    3 )  0.997

Problems Answers: 1. (a) Probability(red) = (b) Probability (white) = (c) Probability (blue) =

(d) Probability (not red) = (e) Probability (red or white) =

=

2. With ball replaced a) Probability (red, white, blue)= Ball not replaced b) Probability (red, white, blue)= 3. Accounting: 100 students Business Statistics: 80 students Students studying both: 30 Probability (student’s studying either accounting or business statistics) : P(A B) = P (A) + P(B) –P (A 5

=

4. (a) Probability ( $18000-$22999) = (b) Probability (less than $23,000) = (c) Probability (either less than $18,000 or atleast $ 40000) =

=

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