1. PROBABILITY AND STATISTICS
WEEK 3
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2. Introduction to Probability
The Classical Interpretation of Probability
Theoretically
The Experimental Interpretation of Probability
Empirically
The Subjective Interpretation of Probability
Subjectively
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3. Sample Space, Experiment, Event
An experiment is any process, real or hypothetical, in which the possible outcomes can be identified ahead of time. An event is a well-defined set of possible outcomes of the experiment.
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4. Recall: Operations of Set Theory
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5. Example
Example: Consider tossing a fair coin. Define the event H as the occurrence of a head. What is the probability of the event H, P(H)?
1. In a single toss of the coin, there are two possible outcomes
2. Since the coin is fair, each outcome (side) should have an equally likely chance of occurring
3. Intuitively, P(H) = 1/2 (the expected relative frequency)
Notes:
This does not mean exactly one head will occur in every two tosses of the coin
In the long run, the proportion of times that a head will occur is approximately 1/2
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6. Experiment Experimental results of tossing a coin 10 times each trial:
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7. Law of Large Numbers
Law of Large Numbers: If the number of times an experiment is repeated is increased, the ratio of the number of successful occurrences to the number of trials will tend to approach the theoretical probability of the outcome for an individual trial
Interpretation: The law of large numbers says: the larger the number of experimental trials n, the closer the empirical probability P(A) is expected to be to the true probability P(A)
In symbols: As

8. Probabilities
Expected value = 1/2
Trial
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9. Axioms and Basic Theorems
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10. mutually exclusive / exhaustive
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11. Example Two dice are cast at the same time in an experiment.
Define the sample space of the experiment.
Find the pairs whose sum is 5 (A) and the pairs whose first die is odd (B).
Are A and B mutually exclusive?
Are A and B exhaustive?
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12. Example In a city, 60% of all households subscribe to the
newspaper A, 80% subscribe newspaper B, and 10% of
all households do not subscribe any newspaper.
If a household is selected at random,
What is the probability that it subscribes to at least one of the two newspapers?
Exactly one of the two newspapers?
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13. Example
a. What is the probability that the individual has a medium auto deductible and a high homeowner’s deductible?
b. What is the probability that the individual has a low auto deductible? A low homeowner’s deductible?
c. What is the probability that the individual is in the same category for both auto and homeowner’s deductibles?
d. What is the probability that the two categories are different?
e. What is the probability that the individual has at least one low deductible level?
f. What is the probability that neither deductible level is low?
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14. Counting Techniques
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15. Permutations Suppose that four-letter words of lower case alphabetic
characters are generated randomly with equally likely outcomes. (Assume that letters may appear repeatedly.)
How many four-letter words are there in the sample space S?
How many four-letter words are there are there in S that start with the letter ”k”?
What is the probability of generating a four-letter word that starts with an ”k” ?
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16. Permutations
How many words of length k can be formed from a set of n (distinct) characters, (where k ≤ n), when letters can be used at most once ?
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17. Permutations
An ordered subset is called a permutation. The number of permutations of size k that can be formed from the n individuals or objects in a group will be denoted by P(n,k).
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18. Permutations Sampling with Replacement. Obtaining Different Numbers.
Birthday Problem?
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19. Combination
Consider a set with n elements. Each subset of size k chosen from this set is called a combination of n elements taken k at a time.
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20. Example
Suppose that a club consists of 25 members and that a president and a secretary are to be chosen from the membership. We shall determine the total possible number of ways in which two people will fill the two positions.
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21. Example
Suppose that a class contains 15 boys and 30 girls, and that 10 students are to be selected at random for a special assignment. We shall determine the probability exactly three boys will be selected.
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22. Example
Suppose that a deck of 52 cards containing four aces is shuffled thoroughly and the cards are then distributed among four players so that each player receives 13 cards. We shall determine the probability that each player will receive one ace.
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23. Example
Suppose that a fair coin is to be tossed 7 times, and it is desired to determine;
the probability p of obtaining exactly three heads
the probability p of obtaining three or fewer heads.
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24. Example
A box containing 3 red, 4 blue and 5 black balls. What’s the probability that drawn 3 balls will be different colours?
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25. Binomial Coefficients
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26. Multinomial Coefficients
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27. Multinomial Coefficients
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28. Example
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29. Example
How many nonnegative integer solutions are there to x + y + z = 17 ?
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30. Example What is the probability the sum is 9 in three rolls of a die ?
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