1. Probability 14.1 Experimental Probability 14.2 Principles of Counting
14.3 Theoretical Probability

2. 14.1
Experimental Probability

3. DEFINITION: EXPERIMENTAL PROBABILITY
Suppose an experiment with a number of possible outcomes is performed repeatedly – say, n times – and that a specific outcome E occurs r times. The experimental probability, denoted by , that E will occur on any given trial of the experiment is given by
Experimental outcomes
Total number or outcomes
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4. THE LAW OF LARGE NUMBERS
If an experiment is performed repeatedly, the experimental probability of a particular outcome more and more closely approximates a fixed number as the number of trials increases.
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5. THE LAW OF LARGE NUMBERS
Ratio of the number of heads to the number of tosses in a coin-tossing experiment:
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6. Example - Computing Experimental Probability from Data
The final examination scores of students in a precalculus class are as shown. Compute the experimental probability that a student chosen at random from the class had a score in the 70s?
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7. Example : continued
Since 6 of the 28 students scored in the 70s, the experimental probability that a student chosen at random had a score in the 70s is
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8. Example: Determining Experimental Probabilities from an Experiment
Christine has five pennies. She is curious how often she should expect to see at most one head when all five coins are flipped onto the floor. To find an answer, she repeatedly flips the five pennies and counts the number of heads that turn up. After repeating the experiment 50 times, she obtains the following frequency table:
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9. Example: continued
On the basis of Christine’s data, what is the experimental probability that a flip of five coins results in at most one head?
The data shows that exactly one head appeared on seven of the trials ad no heads appeared once. This means that the outcome of at most one head occurred = 8 times in the 50 trials, giving the experimental probability of
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10. THE TERMINOLOGY OF PROBABILITY
OUTCOME:
a result of one trial of an experiment
SAMPLE SPACE S:
the set of all outcomes of an experiment
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11. THE TERMINOLOGY OF PROBABILITY
MUTUALLY EXCLUSIVE EVENTS A and B:
two events A and B such that the occurrence of A does not affect its occurrence in B, and vice-versa. In other words they cannot happen at the same time. E.g. Turning left and turning right.
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12. THE TERMINOLOGY OF PROBABILITY
NON-MUTUALLY EXCLUSIVE EVENTS A and B:
If A and B are any two events, they can happen at the same time. E.g. Turning left and scratching your head can happen at the same time.
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13. Example: Computing the Experimental Probability of Non-Mutual Exclusive Events
A penny and a dime are flipped, with the results as shown in the table.
Determine the experimental probability that the dime shows a head or both coins land with the same side up.
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14. EXPERIMENTAL PROBABILITY OF NON-MUTUALLY EXCLUSIVE EVENTS
Let A denote the event “the dime shows a head.” Let B denote the event “both coins land with the same side up.”
and
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15. 14.2
Principle of Counting
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16. PRINCIPLES OF COUNTING
To determine the theoretical probabilities of events, we will need to determine the number of possible outcomes.
Thus, the calculation of theoretical probability is a matter of counting.
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17. DEFINITION: INDEPENDENT EVENTS
Events A and B are independent events if the occurrence or nonoccurrence of event A does not affect the occurrence or nonoccurrence of event B, and vice versa.
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18. ADDITION PRINCIPLE OF COUNTING
If A and B are events, then
and
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19. Example 14.10: Counting and the Word Or
In how many ways can you select a red card or an ace from an ordinary deck of playing cards?
Solution 1: There are 26 red cards in the deck (including two red aces) as well as two black aces. Thus, the desired answer is 28.
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20. Example 14.10: continued and
In how many ways can you select a red card or an ace from an ordinary deck of playing cards?
Solution 2: Let R denote the set of red cards and let A denote the set of aces. Then n(R) = 26, n(A) = 4, n(R ∩ A) = 2, and _
and
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21. ADDITION PRINCIPLE OF COUNTING FOR MUTUALLY EXCLUSIVE EVENTS
If A and B are mutually exclusive events, then
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22. Example 14.12: Choosing a Chocolate
A box of 40 chocolates contains 14 cremes (A), 16 caramels (B), and 10 chocolate-covered nuts. In how many ways can you select a crème or a caramel from the box.
= 30.
Thus, the number of ways of choosing a creme or a caramel is 30.
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23. MULTIPLICATION PRINCIPLE OF COUNTING FOR INDEPENDENT EVENTS
When there are A ways to do one thing,  and B ways to do another, then there are A x B ways of doing both. If A and B are independent events, then the number of ways that A and B can occur in a two-stage experiment is given by
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24. Example 14.14: Counting the Number of Ways to Draw Two Aces
How many ways can two aces be drawn in succession from a deck of ordinary cards
-- With replacement?
If we replace the drawn card, then each
choice is independent, so
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25. -- Without replacement? If we don’t replace the first drawn ace, then
Example 14.14: continued
How many ways can two aces be drawn in succession from a deck of ordinary cards
-- Without replacement?
If we don’t replace the first drawn ace, then
the number of ways to draw a second ace is
impacted. So
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26. Theoretical Probability
14.3
Theoretical Probability
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27. DEFINITION: THEORETICAL PROBABILITY
Let S, the sample space (total outcomes), and let E, an event (desired outcomes), Let n(S) and n(E) denote the number of outcomes in S and E, respectively. The probability of E, denoted by P(E), is given by
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28. DETERMINING PROBABILITY
Determine the probability of obtaining a total score of 3 or 4 on a single throw of two dice.
List the sample space as ordered pairs.
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)
(4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)
(5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)
(6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)
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29. P(white) = desired outcome / total outcomes
Example
Five black marbles, seven white marbles, eight red marbles are placed in an bag. If one is chosen at random, what is the probability of a red marble?
P(white) = desired outcome / total outcomes
P(white) = 8/20 = 2/5 = 0.4 = 40%
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30. PROPERTIES OF PROBABILITY
E.g. Six sided die, the probability of rolling a seven.
E.g. Six sided die, the probability of rolling 1,2,3,4,5, or 6.
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31. PROPERTIES OF PROBABILITY
For any event
E.g. all probabilities are between 0 and 1 and add up to 1 or 100%
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32. Then the odds in favour of A are
DEFINITION: ODDS
Then the odds in favour of A are
favourable outcomes : unfavourable outcomes
Then the odds against A are
unfavourable outcomes : favourable outcomes
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33. Example 14.34: Determining the Odds in Favor of Rolling a 7 or an 11
In the game of craps, one wins on the first roll of
the pair of dice if a 7 or an 11 is thrown. What are
the odds of winning on the first roll?
Let W be the set of outcomes that result 7 or 11.
Since W = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1),
(5,6), (6,5)} and these are 36 ways two dice can
come up.
Thus the odds in favor of W are 8:28 or, 2:7.
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34. DEFINITION: EXPECTED VALUE OF GAMBLING
Average expected payoff in the long run.
positive results = win, negative results =
loss, zero = breakeven.
EV = P(win) x gain – P(lose) x loss
Gain = Winnings – Initial Bet
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35. DEFINITION: EXPECTED VALUE OF GAMBLING
You pay $2 to play the game, if you pull an ace from a deck you win $10, what is the expected value?
EV = (4/52)(8) – (48/52)(2)
EV = –
EV =
Should you play this game? If you play it 100 times what will the results be?
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