Probability

• Quantifying the likelihood that something is going to happen.

• A number from 0 to 1, inclusive– 0 - Impossible

– 1 - Certain, guaranteed

– ½ - a “toss up”

• Can be expressed as a fraction (in lowest terms), decimal, or percent– Usually starts out as a fraction

Probability definition: Event

• An event is one occurrence of the activity whose probability is being calculated.– E.g., we are calculating the probability of dice, an event is

one roll of the dice.

• A simple event cannot be broken down into smaller components– Rolling one dice is a simple event

• A compound event is made up of several simple events– The probability of a compound event is usually a function of

the component simple events. – Rolling two dice is a compound event.

Probability definitions: Outcome, sample space

• An outcome is one possible result of the event.– Rolling a five is one possible outcome of rolling one dice

– Rolling a seven is one possible outcome of rolling two dice

• The sample space is the list of all possible outcomes– One dice: 1, 2, 3, 4, 5, or 6

– Two dice: See next slide

• The size of the sample space is the total number of possible outcomes– One dice: sample space size is 6

– Two dice: sample space size is 36

• A success is an outcome that we want to measure

• A failure is an outcome that we do not want to measure– Failures = Sample space – successes

Two Dice Sample Space

First Die

1 2 3 4 5 6

2nd Die

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

Probability Symbols and Calculation

• The letter P denotes a probability.• Capital letters (A, B, C, etc) represent outcomes• P(A) denotes the probability of outcome A occurring

• Where a success is when outcome A occurs

Number of possible success( )

Size of sample spaceP A

For example: One Dice

• What is the probability of rolling a five with one dice?– Sample space: 1 2 3 4 5 or 6

– Sample space size: 6

– Successful rolls:

– Number of successes:

– P(5) =

• What is the probability of rolling an odd number?– Successful rolls:

– Number of successes:

– P(Prime) =

For example: Two Dice

• What is the probability of rolling a five with one dice?– Sample space size: 36

– Successful rolls:

– Number of successes:

– P(5) =

• What is the probability of rolling a prime number?– Number of successes:

– P(Prime) =

Types of Probability

• Classical– AKA Theoretical or

Empirical

– Events and outcomes in sample space can be determined from the ‘rules of the game’

– E.g., Wheel of fortune

• Geometric– Sample space is some area, a

successful outcome is hitting some target

• Experimental– AKA Relative frequency

– Some activity is observed

– Sample space size is the total number of events observed

– Success is the subset of events in which out outcome occurred

– E.g., basketball toss

Classical probability: Coin flip

• Event: coin flip• Sample space: heads or tails• Sample space size: 2

• Probability of flipping heads • Sucesses:• # of Successes

• P(Heads)

Classical Probability: Cards

• Event: drawing one (or more) cards• Sample space: a deck cards, two colors, each color

has two suits, each suit has 13 ranks deuce to ten, three face cards, ace

• Sample Space size: 52• What is the probability of drawing a 10 of spades?• Successes:• Number of successes:

P(10♠)

Classic Classical Probability: Cards

Successes # of success P

P(Jack)

P(Red)

P(Heart)

Your turn

• From a deck of cards

• P(Face card) =

• P(Red ace) =

• P(6 or less) =

Classical Probability: Collections

• Sample space: a set of items of different characteristics– Sample space size. We will know the total and numbers of each

characteristics

• Event: Picking one (or more) items with a specific characteristics

• E.g., A box of balls: 4 red, 2 blue, 2 green, 2 yellow, 1 white and 1 black.

• Sample size:

• P(red)– Number of successes:

• P(Black or white)– Number of successes:

Your Turn

• If all the tokens we in a bag and picked at random:

• P(Square)• P(2)• P(3 in a triangle)

1

3

2

1

3

2

1

2

3

1

1

2

1

1

1

2

1

1

1

2

3

3

Classic Classical Probability

• Collections with multiple characteristics

• P(North) = • P(Junior) = • P(South upperclassman) =

Frosh Soph Junior Senior

North 400 375 325 350

South 350 300 325 275

Classic Classical Probability

• Collections with multiple characteristics

• P(North) = • P(Junior) = • P(South upperclassman) =

Frosh Soph Junior Senior

North 400 375 325 350

South 350 300 325 275

Classical Probability: Spinner

• Event: Spinning the wheel• Outcome: Spinner stops at a space• Sample space: individual spaces• Sample space size: # of spaces

• P(1)

• P(red) =

• P(Prime)

1

3

2

4

Do now

• A wheel of fortune has 15 spaces and costs 25 cents to play. If you win, you get a $3 prize

• Another wheel has 10 spaces and also costs 25 cents. If you win, you get a prize worth $2.25.

• If you were down to your last 25 cents, which wheel would you play?

• If you had 10 dollars to spend (25 cents at a time), which wheel would you play?

Identifying the events and sample space

• Sometimes we have to enumerate the sample space.

• How many ways are there to arrange the genders of three children?

• Sample space size?

Questions, always questions

• What is the probability of having three girls?

• P(one boy)?

• P(Youngest is a boy)?

• P(At least one boy)?

More types of probability

• Geometric probability• The event is hitting a target on some surface.

Area of the target( )

Area of the surfaceP A

Complimentary events

• If A represents the occurrence of an event, then Ā represents the event not occurring.

• Ā is the compliment of A

• P(Ā) = 1 – P(A)

heads tails

red = black

male = female

redsox = yankees

Odds

• Odds against are the ratio P(Ā):P(A), reduced to lowest terms

• Odds in favor are the reciprocal of the odds against• What are the odds

– Against drawing a red card

– In favor of drawing an ace

– Against rolling a 5

Odds

• Payoff odds against: Net profit : Amount bet• Example: roulette wheel• The payoff odds for picking one number are 35:1

– If you bet $1, you win $35, plus your original bet.– How much do you win if you bet $5?

• What are the actual odds?– 38 spots on the wheel

• Casinos are profitable because the payoff odds are less than the actual odds