Chapter 5 Probability

5.1 Randomness, Probability, Simulation

Simulations

• Read the “1 in 6 wins” game on p. 282-283. This is a great example of a simulation that might help you with the extra credit project.

Probability

• • • Chance behavior (like tossing a coin) is unpredictable in the SHORT RUN but has a predictable pattern in the LONG RUN.

If we toss a coin billions of times, the proportion of heads will eventually approach exactly 0.5. This is called the Law of Large Numbers .

The proportion of times that a specific outcome occurs is called probability.

Probability

• • • Outcome NEVER occurs = 0.

Outcome ALWAYS occurs = 1.

Outcome sometimes occurs = between 0 and 1.

Myths about Randomness

• These are all myths: – Randomness is predictable in the short run.

– The Law of Averages • Some people believe that if they toss a coin 7 times and get all tails, the next one is bound to be a head (because they have to average out eventually) • Future outcomes aren’t necessarily going to make up for an imbalance • Coins, dice, etc. do not have memories!

Simulation

• • • • A model that accurately reflects some other situation Drawing cards to imitate a lottery Rolling a die to imitate buying sodas and looking under the cap Use the State, Plan, Do, Conclude approach on p.289 (this will help with the extra credit)

Chapter 5

5.2 Probability Rules

Probability Models

• • Sample Space- a list of ALL possible outcomes Also include a probability for EACH outcome • • • • • Example: Rolling a die Roll a 1 = 1/6 Roll a 4 = 1/6 Roll a 2 = 1/6 Roll a 5 = 1/6 Roll a 3 = 1/6 Roll a 6 = 1/6 All Probabilities should add up to 1!

Event

• A collection of outcomes • Die Example: Rolling an even number.

• We would add up the probabilities of rolling a 2, 4, or 6 to calculate this.

Basic Rules of Probability

More Rules

• • Complements: The probability that an event DOES NOT occur is equal to 1 – the probability that it DOES occur.

Mutually exclusive (disjoint): events have no outcomes in common. In this case, the probability that one or the other occurs is the sum of the separate probabilities.

**See yellow box on page 302**

CYU answers page 303

1. A person cannot have a cholesterol level of both 240 AND between 200 and 239 at the same time, so they are mutually exclusive.

2. A person has either a cholesterol level of 240 or above or they have a cholesterol level between 200 and 239. P(A or B) = P(A) + P(B) = 0.16 + 0.29 = 0.45

3. P(C) = 1-P(A or B) = 1-0.45 = 0.55

Two-Way Tables

• • • • See page 303-305.

Don’t “double- count” outcomes.

P(A or B) = P(A) + P(B) – P(A and B) This is called the general addition rule.

1.

CYU answers page 305

Heart Non-Heart

Face Card

3 9

Non-Face Card

10 30 2. P (face card and a heart ) = 3/52 3. A and B are NOT mutually exclusive (There are hearts that are also face cards). Probability = 0.423

Venn Diagrams

• • See page 305-308 Notice complements and intersections, unions

Chapter 5

5.3 Conditional Probability and Independence

Conditional Probability

• • • • The word “given” will be used.

Example: What is the probability that a person owns a home, given that they are a high school graduate?

It is denoted P(B|A) “B given A” Definition: The probability that one event happens given that another event is already known to have happened.

CYU answers page 314

1. P(L) = 0.3656. 37% of grades are lower than a B.

2. P(E|L) = 0.219

P(L|E) = 0.50

P(L|E) gives the probability of getting a lower grade if the student is an engineer. We would want to compare that to the probability of getting a lower grade if the student is in liberal arts and social sciences.

Independent Events

• • Events are independent IF the occurrence of one has no effect on the chance of the other one happening.

Events A and B are independent IF P(A|B) = P(A) and… P(B|A ) = P(B) So, we have to compare probabilities to determine independence. See example on page 316.

CYU answers page 317

1. Independent. We are putting the first card back and reshuffling.

2. Not independent. Once we know the suit of the first card, the probability of getting a heart will change.

3. Independent. Once we know the chosen person is a female, this doesn’t tell us if she is right or left handed. 6/7 of students are right handed. Also, 6/7 of women are right handed. Since those are equal, the events are independent.

Tree Diagrams

• • • See page 317 for instructions on drawing a tree diagram. Helps if you know the sample space already.

This type of diagram is good for using the multiplication rule.

General multiplication rule helps us find the probability of events A and B both occurring.

General Multiplication Rule

P(A ∩ B) = P(A) x P(B|A) This rule says that for both of the events to occur, first one must occur. Then, the second one must occur. See example on page 319.

Special Multiplication Rule

• • • • • Different from General Rule P(A) x P(B) Only applies to independent events.

The other rule applies across the board.

This rule is simpler, and we are able to use it when the circumstances are independent.

CYU answers page 323

1. P(one returned safely) = 0.95 Assuming independence, P(safe return on all 20 missions) = (0.95) 20 = 0.3585

2. No. Whether one is a college student and one’s age are NOT independent.

Calculating Conditional Probabilities

• • • See page 324-328 Conditional Probability Formula (p.324) See example on page 327 about screening the hearing of babies.

Tips

• • • Most conditional probabilities can be solved using a tree diagram or two way table.

Tree diagrams are easier most of the time if conditional probabilities are given.

Two way tables may be easier when counts or proportions are given.