6.1 Simulation Probability is the branch of math that describes the pattern of chance outcomes It is an idealization based on imagining what would happen in an infinitely long series of trials. Probability calculations are the basis for inference Probability model: We develop this based on actual observations of a random phenomenon we are interested in; use this to simulate (or imitate) a number of repetitions of the procedure in order to calculate probabilities (Example 6.2, p. 393)

Simulation Steps State the problem or describe the random phenomenon. State the assumptions. Assign digits to represent outcomes. Simulate many repetitions. State your conclusions.

State the problem or describe the random phenomenon (above). Ex: Toss a coin 10 times. What’s the likelihood of a run of at least 3 consecutive heads or 3 consecutive tails? State the problem or describe the random phenomenon (above). State the assumptions. Assign digits to represent outcomes. Simulate many repetitions. State your conclusions.

6.2 Probability Models Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run! Random is not the same as haphazard! It’s a description of a kind of order that emerges in the long run. The idea of probability is empirical. It is based on observation rather than theorizing = you must observe trials in order to pin down a probability! The relative frequencies of random phenomena seem to settle down to fixed values in the long run. Ex: Coin tosses; relative frequency of heads is erratic in 2 or 10 tosses, but gets stable after several thousand tosses!

Example of probability theory (and its uses) Tossing dice, dealing cards, spinning a roulette wheel are all examples of deliberate randomization Describing… The flow of traffic A telephone interchange The genetic makeup of populations Energy states of subatomic particles The spread of epidemics Rate of return on risky investments

Exploring Randomness You must have a long series of independent trials. The idea of probability is empirical (need to observe real-world examples) Computer simulations are useful (to get several thousands of trials in order to pin down probability)

Sample space for trails involving flipping a coin = Sample space for rolling a die = Probability model for flipping a coin = Probability model for rolling a die =

Event 1: Flipping a coin Event 2: Rolling a die 1) How many outcomes are there? List the sample space. Tree diagram: Multiplication Rule 2) Find the probability of flipping a head and rolling a 3: Find the probability of flipping a tail and rolling a 6:

… If you were going to roll a die, pick a letter of the alphabet, use a single number and flip a coin, how many outcomes could you have? 2) As it relates to the experiment above, define an event and give an example:

Sample space as an organized list Flip a coin four times. Find the sample space, then calculate the following: P(0 heads) P(1 head) P(2 heads) P(3 heads)

Sampling with replacement: If you draw from the original sample and put back whatever you draw out Sampling without replacement: If you draw from the original sample and do not put back whatever you drew out! EXAMPLE: Find the probability of getting one ace, then another ace without replacement. Find the probability of getting one ace, then another ace with replacement.