6.1 Simulation

2. 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)

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

4. 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.

5. 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!

6. 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

7. 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)

8. 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 =

10. 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:

11. … • 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:

12. 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)

14. 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.