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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