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
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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: