Chapter Five Notes

Probabilities are always between _____ and _____.

0 ; 1

Simulation steps example: Do

1) yes 2) yes 3) no 4) no 5) yes 6) yes 7) yes 8) yes

Five Basic Probability Rules

1. For any event A, the probability of A occurring, P(A) is between 0 and 1. 2. If S is the sample space in a probability model, then P(S)=1. The sum of all probabilities in S is 1. 3. In the case of equally likely outcomes, P(A)= # outcomes of event A/total # of possible outcomes 4. Compliment rule P(A^c)= 1-P(A) 5. Addition rule ONLY if events are mutually exclusive P(A or B)= P(A)+P(B)

Simulation steps

1. State 2. Plan 3. Do 4. Conclude

Simulation steps example: Conclude

Based on our simulation, there is a 6/8 ~ .75 chance of getting 3 or more consecutive heads/tails in 10 coin flips.

Which of the following outcomes from a coin flipped 6 times is more probable? HTHTTH TTTHHH

Because each flip has a 1/2 chance of getting heads/tails, and chance processes have no memory, the outcomes are equally probable.

Choose one person at random from a group of which 73% are employed.

Employed: 01-73 OR 00-72 Unemployed: 74-99, 00 OR 73-99

Choose a person at random from a group of which 50% are employed, 20% are unemployed, and 30% are not in the labor force.

Employed: 1-5 OR 0-4 Unemployed: 6-7 OR 5-6 Not employed: 8-9, 0 OR 7-9

In a group of 50 students (20 girls and 30 boys), we want to see the probability that 5 girls are selected.

Girls: 01-20 Boys: 21-50 Ignore: 51-99, 00

Imagine flipping a fair coin three times. Describe the probability model for this chance process and use it to find the probability of getting at least 1 head in three flips.

HHH HTH THH HHT HTT TTH THT TTT Since each outcome is equally likely, they each have a probability of ~1/8. P(getting at least one head in 3 flips)= 7/8

The probability of getting a sum of 7 when rolling two dice is 1/6. Interpret this value.

If we roll 2 dice many times, the probability of getting a sum of 7 is ~1/6.

Simulation steps example: Plan

Let 0-4 be heads and 5-9 be tails. Read 1 digit at a time until you've looked at 10 numbers. Repeat numbers are okay. Record if a streak of 3 or more heads/tails occurs.

How do you interpret a probability?

Long-run (many trials) relative frequency (percent)

Example: Streakiness Suppose that a basketball announces suggests that a certain player is streaky. That is, the announcer believes that if the player makes a shot, then he is more likely to make his next shot. As evidence, he points to a recent game where the player took 30 shots and had a streak of 7 made shots in a row. Is this convincing evidence of streakiness or could it have occurred simply by chance? Assuming this player makes 48% of his shots and the results of a shot don't depend on previous shots, how likely is it for the player to have a streak of 7 or more made shots in a row?

State: probability of this player getting 7 or more shots out of 30 Plan: 01-48= makes shot ; 49-99, 00= misses shot. Use randInt(00, 99, 30). Repeat #s are okay. Each trial, record if a streak of 7+ shots were made.

Law of averages

TTTTTT >"Heads is more likely for next flip" MYTH! NOT true!

What does it mean if two events are mutually exclusive, a.k.a. disjoint events?

The events cannot happen at the same time; P(A AND B)= 0 i.e. Rolling a die: get a number greater than 3 and a number less than 3.

What is the purpose of simulations?

To estimate probabilities when theoretical probabilities are too difficult to calculate

Simulation steps example: State

Tossing a coin 10 times, what is the likelihood of a run of at least 3 consecutive heads or 3 consecutive tails?

An __________ is any collection of outcomes from some chance process. That is, it is a __________ of the sample space.

event; subset

The __________ says that as you perform more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value.

law of large numbers

WITHOUT REPLACEMENT: After an individual has been selected, it can no longer be selected again until the next simulation. In simulations, repeating numbers are __________.

not allowed *often when using #s to represent physical objects

WITH REPLACEMENT: Each individual has an equal choice of being selected each time. In simulations, repeating numbers are __________.

okay *often when using #s to represent %s

A __________ is a description of some chance process that consists of two parts: a __________ and the __________ for each outcome.

probability model; sample space; probability

The __________, S, of a chance process is the set of all possible outcomes.

sample space