Statistics 4.1: Probabilities

Probability Rule #1

*O < P(E) < 1* The probability of an event must be a number between 0 and 1, inclusive. *If an event must happen, P(E) = 1* *If an event is impossible, P(E) = 0* This is always the range!! All probabilities should be written as reduced fractions, decimals rounded to 3 non-zero place values, or as a percent. Ex. 1/3, 0.333, or 33.3%

Event

*Subset* of the sample space.

What are the 3 types of probabilities?

1. *Classical Probability* assumes that every outcome has an equal chance of occurring. Ex. Tossing one die. Each side has an equal chance of turning up. 2. *Imperical Probability* are computed by experimentation. Ex. The guy who wanted to protect the endangered tortoise from getting ran over... (what? lol) 3. *Subjective Probability* is based upon a personal experience. Ex. Assuming you might get pinched on St. Patrick's day. *NOTE: We will only be working on Classical Probability!!*

Example

Example: Toss 1 die. Sample space: {1, 2, 3, 4, 5, 6} Event: Roll a prime number: {2, 3, 5} Probability: P(E) = 3/6 = *1/2*

Suggestions for Study

I'd maybe go over the dice problems, and remember to look at any starred problems on the HW, because a star = that it was a difficult problem.

You should memorize these denominators for the card problems.

If it asks about how many Kings, Queens, Jacks, or Aces, it should be: 4/52 If it asks about how many Clubs, Spades, Diamonds, or Hearts, it should be: 13/52 If it asks about any specific number such as an 8, 11, ect., it should be: 4/52 If it asks about how many reds or blacks, it should be: 26/52

Example 2 (Cont'd)

Okay, so I don't know if this makes sense, but I just wanted to remind you that when you're using that chart to figure out 2-dice problems, you have the sum column and the frequency column. When a problem asks you something like this: What is the probability of getting a sum that is less than 5 and even? What you do is look at the sum column. You'll find that there are two numbers less than 5 that are even (4 and 2). So what you do is take the corresponding frequencies of those numbers (which are 3 and 1), you add them up, and then put them over the total number of outcomes, which is of course, 36. So, the answer wold be p = 4/36 = *1/9* You look at the question, find the numbers you need to on the sum side, and then take the corresponding frequency and put it over the total number of outcomes. BUT, when it asks you about the probability of getting doubles, it's a little bit different... In this case, you just count how many ways you can get doubles, which is 6, and put that number over 36.

Probability Rule #2

P(E) + P(E bar) = 1 P(E) = 1 - P(E bar) *P(E bar) = 1 - P(E)* <-- most important step. The probability that E does not occur is P(E bar) = 1 - P(E)

Probability Equation

P(E) = # of outcomes in E / # of outcomes in the sample space

Outcomes

Probability is a process that leads to *well-defined results called outcomes*. Ex. Tossing a die, flipping a coin.

Tree Diagrams

Process that helps list all the outcomes for a sequence of events. It consists of the little dot thing called a node, which represents a decision. It also involves branches, which represent the outcome. See your notes for different examples.

Probability

The chance that an event will occur.

What is the complement of an event?

The complement of event E is denoted as *E bar*, and it consists of all the outcomes in the sample space that are not in E. Ex. If you toss one die {1, 2, 3, 4, 5, 6} E = Get a prime #, E bar would be don't get a prime # E = {2, 3, 5}, E bar would be {1, 4, 6}

Sample Space

The set of *all possible outcomes* of the experiment. Sometimes it won't be possible to do this because there's too many outcomes.

Example 3

When you are dealing with card deck problems, you also have to create a special table (see your notes). But, remember that there are 52 cards in a standard deck. Select 1 card from a standard deck. Compute the following possibilities. a) Get a king: p = 4/52 = *1/13* *Remember, there is a King of each type... Hearts, diamonds, spades, and clubs.* b) Get an 8 *or* a heart: p = 4/52 + 12/52 = *4/13* *TREAT "OR" LIKE A + SIGN!!* c) Get a Jack and a Black card: p = 2/52 = *1/26* *THE WORD "AND" IS WHAT THE TWO HAVE TO HAVE IN COMMON* d) Get a face card: p = 12/52 = *3/13* *A face card include Jack, Queen, King... NOT ACE.*

Example 2

You have *2 dice*, there are 36 outcomes, because *6 (1 die) x 6 (1 die) = 36*. For problems that involve *2 dice*, you have to *create a special chart* (see your notes.. *MEMORIZE THE CHART!*). Assume that 2 dice are tossed. Compute the following possibilities. a) Get a sum of 9: p = 4/36 = *1/9* *b)* Get doubles: 6/36 = *1/6* (BE CAREFUL WITH THIS ONE! Think about how the dice look, okay? For instance, if you roll snake eyes, the sum of the dice will be 2, and there will be a dot on each of them. BUT, if you get a sum of 3, that means that there has to be a dot on one of them and two dots on the other... Thus, that's not doubles.) c) Get a sum that is less than 5 and even: p = 4/36 = *1/9* d) Get 36: p = *0* (because the event is impossible!) e) Get a number greater than 8 and prime: p = 2/36 = *1/18*

Example 4

You have a jar of jelly beans. 20 red, 15 green, 10 purple, 4 black, 21 pink. Let's say you select one jelly bean... Compute the following probabilities. a) It is red: p = 20/70 = *.286* b) It is not purple: p = 60/70 = *.857* or, you can do it like this: p = 1 - 10/70 = .143 p = 1 - .143 = .857 c) It is orange: p = 0/70 = *0* d) It is not green or pink: p = 34/70 = *.486* e) It is red or black: p = 24/70 = *.343*