6.1 Simulation6.1 Simulation ProbabilityProbability is the branch of math that describes the is the branch of math that describes the

pattern of chance outcomespattern of chance outcomes ProbabilityProbability is an idealization based on imagining what is an idealization based on imagining what

would happen in an infinitely long series of trials.would happen in an infinitely long series of trials. Probability calculations are the basis for inferenceProbability calculations are the basis for inference Probability modelProbability model: We develop this based on actual : We develop this based on actual

observations of a random phenomenon we are observations of a random phenomenon we are interested in; use this to simulate (or imitate) a number interested in; use this to simulate (or imitate) a number of repetitions of the procedure in order to calculate of repetitions of the procedure in order to calculate probabilities (Example 6.2, p. 393)probabilities (Example 6.2, p. 393)

Simulation StepsSimulation Steps

1)1) State the problem or describe the State the problem or describe the random phenomenon.random phenomenon.

2)2) State the assumptions.State the assumptions.3)3) Assign digits to represent outcomes.Assign digits to represent outcomes.4)4) Simulate many repetitions.Simulate many repetitions.5)5) State your conclusions.State your conclusions.

ExEx: Toss a coin 10 times. What’s : Toss a coin 10 times. What’s the likelihood of a run of at least 3 the likelihood of a run of at least 3 consecutive heads or 3 consecutive consecutive heads or 3 consecutive tails?tails?

1)1) State the problem or describe the State the problem or describe the random phenomenon (above).random phenomenon (above).

2)2) State the assumptions.State the assumptions.3)3) Assign digits to represent outcomes.Assign digits to represent outcomes.4)4) Simulate many repetitions.Simulate many repetitions.5)5) State your conclusions.State your conclusions.

6.2 Probability Models6.2 Probability Models Chance behaviorChance behavior is unpredictable in the short run but has is unpredictable in the short run but has

a regular and predictable pattern in the long run!a regular and predictable pattern in the long run! RandomRandom is not the same as haphazard! It’s a description is not the same as haphazard! It’s a description

of a kind of order that emerges in the long run.of a kind of order that emerges in the long run. The idea of probability is The idea of probability is empirical.empirical. It is based on It is based on

observation rather than theorizing = you must observe observation rather than theorizing = you must observe trials in order to pin down a probability!trials in order to pin down a probability!

The The relative frequenciesrelative frequencies of random phenomena seem to of random phenomena seem to settle down to fixed values in the long run.settle down to fixed values in the long run. Ex: Coin tosses; relative frequency of heads is erratic Ex: Coin tosses; relative frequency of heads is erratic

in 2 or 10 tosses, but gets stable after several in 2 or 10 tosses, but gets stable after several thousand tosses!thousand tosses!

Example of probability Example of probability theory (and its uses)theory (and its uses)

Tossing dice, dealing cards, spinning a roulette Tossing dice, dealing cards, spinning a roulette wheel (exs of deliberate randomization)wheel (exs of deliberate randomization)

Describing…Describing…The flow of trafficThe flow of trafficA telephone interchangeA telephone interchangeThe genetic makeup of populationsThe genetic makeup of populationsEnergy states of subatomic particlesEnergy states of subatomic particlesThe spread of epidemicsThe spread of epidemicsRate of return on risky investmentsRate of return on risky investments

Exploring RandomnessExploring Randomness1)1) You must have a long series of You must have a long series of independent independent

trials.trials.2)2) The idea of probability is empirical (need to The idea of probability is empirical (need to

observe real-world examples)observe real-world examples)3)3) Computer simulations are useful (to get Computer simulations are useful (to get

several thousand of trials in order to pin down several thousand of trials in order to pin down probability) probability)

Sample space for trails involving flipping Sample space for trails involving flipping a coin = ?a coin = ?

Sample space for rolling a die = ?Sample space for rolling a die = ? Probability model for flipping a coin =?Probability model for flipping a coin =? Probability model for rolling a die = ?Probability model for rolling a die = ?

Event 1: Flipping a coinEvent 1: Flipping a coinEvent 2: Rolling a dieEvent 2: Rolling a die1) How many outcomes are there? List the 1) How many outcomes are there? List the sample space. sample space. Tree diagram: * RuleTree diagram: * Rule2) Find the probability of flipping a head and 2) Find the probability of flipping a head and rolling a 3. Find the probability of flipping a rolling a 3. Find the probability of flipping a tail and rolling a 6.tail and rolling a 6.3) # of outcomes?3) # of outcomes?

……

1) If you were going to roll a die, pick a 1) If you were going to roll a die, pick a letter of the alphabet, use a single letter of the alphabet, use a single number and flip a coin, how many number and flip a coin, how many outcomes could you have?outcomes could you have?

2) As it relates to the experiment above, 2) As it relates to the experiment above, define an event and give an example:define an event and give an example:

Sample space as an Sample space as an organized listorganized list

Flip a coin four times. Find the sample Flip a coin four times. Find the sample space, then calculate the following:space, then calculate the following:

1)1) P(0 heads)P(0 heads)2)2) P(1 head)P(1 head)3)3) P(2 heads)P(2 heads)4)4) P(3 heads)P(3 heads)

Sampling with replacementSampling with replacement: If you draw from : If you draw from the original sample and put back whatever the original sample and put back whatever you draw out you draw out

Sampling without replacementSampling without replacement: If you draw : If you draw from the original sample and from the original sample and do notdo not put back put back whatever you drew out!whatever you drew out!

EXAMPLE: EXAMPLE: 1)1) Find the probability of getting one ace, then 2 Find the probability of getting one ace, then 2

aces without replacement.aces without replacement.2)2) Find the probability of getting one ace, then 2 Find the probability of getting one ace, then 2

aces with replacement.aces with replacement.