Question

QUESTION 2 a) Discuss, briefly, the three ways of measuring the probability of an event. [3 Marks b) Alhassan and Bawa are planning to form a partnership to build and operate a number of “car-wash bases" and that there Alhassan feels that there is a fifty-fifty chance that any new unit will show a profit for the first year. Bawa feels that the odds are 2 to 1. Due to Alhassan's experience, he is three times as likely to be right as Bawa. Suppose the partnership came into being and out of the six units opened, five of them showed a profit for the first year and the sixth showed a loss. i) Calculate Alhassan's and Bawa's corresponding posterior probabilities. [6 Marks ii) How does the result in i) affect Alhassan and Bawa's claim? [1 Mark]

Answer 1

Three Types of Probability

1. Classical:

(equally probable outcomes) Let S=sample space (set of all possible distinct outcomes). Then the probability of an event =number of ways the events occur/number of outcomes S

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 Definition

The probability of an event in an experiment is the proportion (or fraction) of times the event occurs in a very long (theoretically infinite) series of (independent) repetitions of experiment. (e.g. probability of heads=0.4992)

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: P(A) = lim n→∞ nA n . 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).

Example 1: Roll of a Die S = {1, 2, · · · , 6} Probabilities: Roll the given die 100 times (say) and suppose 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 Probability

The probability of an event is a "best guess" by a person making the statement of the chances that the event will happen. (e.g. 30% chance of rain)

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.

Example 1: Horse Race Consider a horse race with 8 horses running. What is the probability for a particular horse to win? Is it reasonable to assume that the probability is 1/8? Note that we can’t apply the relative-frequency approach. People regularly place bets on the outcomes of such “onetime” experiments based on their judgment as to how likely it is for a particular horse to win. Indeed, having different judgments is what makes betting possible!