MBA 604 Introduction Probaility and Statistics Lecture Notes

Probabilistic Law. In a fair coin tossing experiment the percentage of (H)eads is very
close to 0.5. In the model (abstraction): P (H) =0.5 exactly.
Why Probabilistic Reasoning?
Example. Toss 5 coins repeatedly and write down the number of heads observed in each
trial. Now, what percentage of trials produce 2 Heads?
answer. Use the Binomial law to show that
5
P (2Heads) = (0.5)
2
2 (1 − .5) 3
= 5!
2!3! (0.5)2 (.5) 3 =0.3125
Conclusion. There is no need to carry out this experiment to answer the question.
(Thus saving time and effort).
2. The Interplay Between Probability and Statistics. (Theory versus Application)
(i) Theory is an exact discipline developed from logically defined axioms (conditions).
(ii) Theory is related to physical phenomena only in inexact terms (i.e. approximately).
(iii) When theory is applied to real problems, it works ( i.e. it makes sense).
Example. A fair die is tossed for a very large number of times. It was observed that
face 6 appeared 1, 500. Estimate how many times the die is tossed.
Answer. 9000 times.
Review Exercises: Probability
Please show all work. No credit for a correct final answer without a valid argument.
Use the formula, substitution, answer method whenever possible. Show your work
graphically in all relevant questions.
1. An experiment consists of tossing 3 fair coins.
(i) List all the elements in the sample space.
(ii) Describe the following events:
A = { observe exactly two heads}
B = { Observeatmostonetail}
C = { Observe at least two heads}
D = {Observe exactly one tail}
(iii) Find the probabilities of events A, B, C, D.
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