Transcript Probability model

6.1 Simulation
 Probability is the branch of math that describes the
pattern of chance outcomes
 Probability 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
1) State the problem or describe the
random phenomenon.
2) State the assumptions.
3) Assign digits to represent outcomes.
4) Simulate many repetitions.
5) 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?
1) State the problem or describe the
random phenomenon (above).
2) State the assumptions.
3) Assign digits to represent outcomes.
4) Simulate many repetitions.
5) 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 (exs 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
1) You must have a long series of independent
trials.
2) The idea of probability is empirical (need to
observe real-world examples)
3) Computer simulations are useful (to get
several thousand 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: * Rule
2) Find the probability of flipping a head and
rolling a 3. Find the probability of flipping a tail
and rolling a 6.
3) # of outcomes?
…
1) 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:
1) P(0 heads)
2) P(1 head)
3) P(2 heads)
4) 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:
1) Find the probability of getting one ace, then 2
aces without replacement.
2) Find the probability of getting one ace, then 2
aces with replacement.