1. Unit 3: Probability 3.1: Introduction to Probability

2. Definitions Fair game Experiment Trial
All players have an equal chance of winning
Each player can expect to win or lose the same number of times in the long run
Experiment
Has a well-defined outcome
E.g. tossing a coin 20 times
Drawing a card from a deck and replacing it 35 times
Trial
One repetition of an experiment
E.g. flipping a coin once
Drawing one card from a deck

3. Definitions Discrete random variable Expected value Simulation
A variable that assumes a unique value for each outcome
Expected value
The value the mean of the random variable value tends towards after many repetitions
Simulation
An experiment that models an actual event
Simple Event
An event where there is only 1 outcome

4. Probabilities There are three kinds of probabilities Experimental
Based on observation
Also called empirical or relative frequency probability
Theoretical
Based on mathematical analysis
Also called classical or a priori probability
Subjective
Estimate based on informed guesswork

5. Theoretical Probability
Sample Space (S)
Consists of all possible outcomes of an experiment
Event Space (A)
- Consists of all outcomes that correspond to
the event of interest S A

6. Theoretical Probability
n(S)
The number of elements in set S
n(A)
The number of elements in set A
The probability of an event A is

7. Theoretical Probability P(A)
Likelihood that an event will occur
0 ≤ P(A) ≤ 1
P(A) = 0
Impossible event
P(A) = 1
Event is a certainty
P(A) < 0 or P(A) > 1 have no meaning
Probability can be expressed as a percent, decimal, or fraction

8. Important Basic Info: Coins
Coins have two faces
Heads (H)
Tails (T)
A fair coin has equal likelihood of landing heads or tails

9. Important Basic Info: Dice
Singular = die
For a fair die, each side has equal likelihood of landing face-up
A six-sided die has 6 sides: 1, 2, 3, 4, 5, 6
An eight-sided die has 8 sides: 1-8
A dodecahedral die has sides 12
Source:

10. Important Basic Info: Cards
52 cards in a deck
4 suits, 2 colours:
spades ♠, clubs ♣, hearts ♥, diamonds ♦
13 values in each suit:
A(ce), 2, 3, 4, 5, 6, 7, 8, 9, 10, J(ack), Q(ueen), K(ing)
Face cards: Jack, Queen, King
Some decks may contain Jokers (2)
Source:

11. Example 1: Rolling a Die
Find the probability that a single roll of a die will result in a number less than 4.
A = {1, 2, 3}
Therefore the probability of rolling a number less than 4 is
= 0.5

12. Example 2: Drawing a Card at Random
A card is drawn at random from an ordinary deck of 52 playing cards. What is the probability of drawing a king?
A = {K, K, K, K}
Therefore the probability of drawing a king is
= 0.077

13. Complementary Events S A' A
The complement of an event A is given by A′
Means that the event does not occur
E.g. A = die rolls a 3
A′ = die rolls anything other than a 3
A and A′ together will include all possible outcomes
Sum of their probabilities must be 1
P(A) + P(A′) = 1
P(A′) = 1 – P(A) S A A'

14. Example 4
What is the probability that a randomly drawn integer between 1 and 40 is not a perfect square?
A = set of perfect squares between 1 and 40
= {1, 4, 9, 16, 25, 36}
A′ = not a perfect square
Therefore the probability that a randomly drawn integer between 1 and 40 is not a perfect square is 0.85

15. #5a on HO
Each of the letters of the word PROBABILITY is printed on same-sized pieces of paper and placed in a bag. The bag is shaken and one piece of paper is drawn. (Consider Y as a vowel.)
A) What is the probability that the letter A is selected?
A = letter A is drawn, S = number of letters
there is 1 A and there are 11 letters
Therefore the probability of drawing a letter A is
= 0.09

16. #5b on HO What is the probability that the letter B is selected?
A = letter B is drawn
Therefore the probability of drawing a letter B is
= 0.18

17. #5c on HO What is the probability that a vowel is selected?
Therefore the probability of drawing a vowel is
= 0.45

18. #5d on HO What is the probability that a consonant is not selected?
A = consonant selected
A′ = consonant not selected
Note: this is the same as saying P(vowel selected)
Therefore the probability of not drawing a consonant is 0.45.

19. FND: pg. 218 #1-15