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