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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.

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 coinEvent 2: Rolling a die1) How many outcomes are there? List the sample space. Tree diagram: Multiplication Rule2) 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 notput 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.

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