Unit 3: Probability3.1: Introduction to Probability
Definitions• Fair game
– 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
Definitions
• Discrete random variable– 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
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
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
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
n( )( )
n( )
AP A
S
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
Important Basic Info: Coins
• Coins have two faces– Heads (H)– Tails (T)
• A fair coin has equal likelihood of landing heads or tails
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
Source: http://www.oitc.com/Dice/images/Dice.gif
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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: http://david.bellot.free.fr/svg-cards/images/svg-cards-2.0.jpg
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}n( )
( )n( )
AP A
S
3
6
1
2
Therefore the probability of rolling a number less than 4 is 1
2= 0.5
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}n( )
( )n( )
AP A
S
4
52
1
13
Therefore the probability of drawing a king is 1
13= 0.077
Complementary Events• The complement of an event A is given by A′• Means that the event does not occur
– E.g. A = die rolls a 3A′ = 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)
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( ') 1 ( )
n( )1
n( )
P A P A
A
S
61
40
3( ') 1
20P A
17
20
0.85
Therefore the probability that a randomly drawn integer between 1 and 40 is not a perfect square is 0.85
#5a on HOEach 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?
Therefore the probability of drawing a letter A is
•A = letter A is drawn, S = number of letters•there is 1 A and there are 11 letters
n( )( )
n( )
AP A
S
1
11 1
11= 0.09
#5b on HO
What is the probability that the letter B is selected?
Therefore the probability of drawing a letter B is
•A = letter B is drawn
2
11
n( )( )
n( )
AP A
S
2
11= 0.18
#5c on HO
• What is the probability that a vowel is selected?
Therefore the probability of drawing a vowel is
•A = vowel is selected
5
11
5
11
n( )( )
n( )
AP A
S
= 0.45
#5d on HO
• What is the probability that a consonant is not selected?
Therefore the probability of not drawing a consonant is 0.45.
Note: this is the same as saying P(vowel selected)
•A = consonant selected•A′ = consonant not selected( ') 1 ( )
n( )1
n( )
P A P A
A
S
61
11
5
11
• FND: pg. 218 #1-15