Probability

= Everything we ever needed to know for the AP Statistics exam about: = Probability! media type="youtube" key="aK4-F_4EzwU" height="268" width="336"

Definitions
** Conditional Probability - an e vent B in relationship to an event A is the probability that event B occurs given that event A has already occurred. The notation for conditional probability is P(B|A), read as the probability of B given A. ** Continuous random variable - takes all values in an interval of numbers, area under the curve (density curves) Discrete random variable -has a countable number of possible values, can be an exact number, (sum=1) Empty Event - an event with no outcomes(an impossible event) Event - set of outcomes of a random phenomenon Expected value -the sum of the products obtained by multiplying each value X by the corresponding probability P.  Independent - no outcomes in common Independent Trials - one trial must NOT influence the outcome of any other Intersection - of any collection of events is the event that all of the events will occur  Law of Large Numbers - the actual mean (x bar) gets closer to the distribution mean (mu) as more trials are made Probability - the proportion of times the outcome would occur (long term relative frequency) Probability Distribution -for a discrete variable, is a listing or formula giving the probability for each value of the random variable Probability Model - random phenomenon consisting of sample space and a way to assign probabilities Random - if outcomes are uncertain but there is a pattern if a large number of repetitions Random variable - variable with a number value of a random phenomenon Sample Space(s) - set of ALL possible outcomes

Rules of Probability
 >  >    ** For Mutually Exclusive Probabilities... **
 * Probability must be a number between 0 and 1 inclusive.
 * All possible outcomes must have a total prob=1 (sample space=1)
 * "or" means add ex. the probability of a 6 or an even number on a die=(1/6)+(3/6)=(4/6)=(2/3)
 * "and" means multiply ex. probability of a girl and blonde=P(girl)*P(blonde)
 * Union means the same as "or"--> ADD
 * <span style="font-family: 'Comic Sans MS',cursive;">Intersect means the same as "and"-->MULTIPLY
 * <span style="font-family: 'Comic Sans MS',cursive;">To prove independence: P(AnB) = P(A)xP(B)
 * <span style="color: #ff00ff; font-family: 'Comic Sans MS',cursive; font-size: 120%;">For Independent Probabilities... **
 * <span style="font-family: 'Comic Sans MS',cursive;">2 events are independent if knowing 1 event doesn't change the probability that the other occurs
 * <span style="font-family: 'Comic Sans MS',cursive;">ex: putting cards back in a deck after drawing
 * <span style="font-family: 'Comic Sans MS',cursive;"> Probability (AnB) = P(A)(B)
 * <span style="font-family: 'Comic Sans MS',cursive;">Probability(AuB) = P(A) + P(B)- P(A)(P(B))
 * <span style="color: #ff00ff; font-family: 'Comic Sans MS',cursive; font-size: 120%;">For Dependent Probabilities... **
 * <span style="font-family: 'Comic Sans MS',cursive;">Probability(A/B)= P(AnB) / P(B)
 * <span style="font-family: 'Comic Sans MS',cursive;">Probability(B/A)= P(AnB) / P(A)
 * <span style="font-family: 'Comic Sans MS',cursive; line-height: 15px;">Probability (Not A)= P(1-A)


 * <span style="font-family: 'Comic Sans MS',cursive; line-height: 15px;">Probability(A) = P(1-Not A)

<span style="font-family: 'Comic Sans MS',cursive;">
 * <span style="font-family: 'Comic Sans MS',cursive; line-height: 15px;">Probability (AuB) = P(A+B)

<span style="color: #00ff00; font-family: 'Comic Sans MS',cursive; font-size: 120%;">Probability in the Calculator
<span style="color: #9900ff; font-family: 'Comic Sans MS',cursive;"> **(they must be binomial to use these *remember BINS (bi, independent, number, probability)*)**

<span style="font-family: Tahoma,Geneva,sans-serif;">**<span style="color: #9900ff; font-family: 'Comic Sans MS',cursive;">*Note: ** **<span style="color: #9900ff; font-family: 'Comic Sans MS',cursive;">Each trial MUST be independent to use these functions.*

****<span style="color: #9900ff; font-family: 'Comic Sans MS',cursive;">binompdf ** **<span style="color: #000000; font-family: 'Comic Sans MS',cursive; font-weight: normal;">will find probability at a specific number ** <span style="font-family: 'Comic Sans MS',cursive;"> in calc--> (total, prob, # of successes) ex. Each time a mouse pushes a lever it has a .3 chance of getting a food pellet. It presses the lever 30 times. What is the probability that the mouse gets 15 pellets. binompdf(30, .3, 15)

in calc--> (total, prob, # of successes) with "at least"--> **<span style="font-family: 'Comic Sans MS',cursive;">1- **<span style="font-family: 'Comic Sans MS',cursive;">binomcdf(total, prob, **<span style="font-family: 'Comic Sans MS',cursive;"># of successes -1 **<span style="font-family: 'Comic Sans MS',cursive;">)ex. Each time a mouse pushes a lever it has a .3 chance of getting a food pellet. It presses the lever 30 times. What is the probability the mouse gets AT MOST 15 pellets. binomcdf(30, .3, 15) ex. Each time a mouse pushes a lever it has a .3 chance of getting a food pellet. It presses the lever 30 times. What is the probability the mouse gets AT LEAST 15 pellets. 1-binomcdf(30, .3, 14)
 * <span style="color: #9900ff; font-family: 'Comic Sans MS',cursive;">binomcdf **<span style="font-family: 'Comic Sans MS',cursive;"> will cumulate down to 0 (at most)

<span style="color: #00ff00; font-family: 'Comic Sans MS',cursive; font-size: 130%;">Tree Diagrams
Method of showing sample space (or list all possibilities)

<span style="font-family: 'Comic Sans MS',cursive;"> A bag contains 3 black balls and 5 white balls. Paul picks a ball at random from the bag and replaces it back in the bag. He mixes the balls in the bag and then picks another ball at random from the bag. a) Construct a probability tree of the problem. b) Calculate the probability that Paul picks: <span style="display: block; font-family: 'Comic Sans MS',cursive; text-align: left;"> i) two black balls ii) a black ball in his second draw
 * Example: **

<span style="font-family: Times,helvetica,sans-serif; font-size: medium; line-height: normal;">
 * Solution: **

<span style="color: #00ff00; font-family: 'Comic Sans MS',cursive; font-size: 130%;">Empirical Rule


<span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 130.13%; text-align: left;">**<span style="color: #00ff00; font-family: 'Comic Sans MS',cursive; font-size: 130%;">Z-Scores ** <span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 100.1%; text-align: left;"> AKA standardized scores

z= (number-mean)/(standard deviation)

Used with NORMAL data

After finding the z-score, use the z-score chart to find the probability. The chart reads to the left, so in cases of "greater than" a certain number, use one minus the probability found in the chart.

= Probability Practice!! =

<span style="color: #000000; font-family: 'Comic Sans MS',cursive; font-size: 112%; font-weight: normal;">A local arcade is hosting a tournament in which contestants play an arcade game with possible scores ranging from 0 to 20. The arcade has set up multiple game tables so that all contestants can play the game at the same time; thus contestant scores are independent. Each contestant's score will be recorded as he or sshe finishes, and the contestant with the highest score it the winner. <span style="color: #000000; display: block; font-family: 'Comic Sans MS',cursive; font-size: 123.2%; font-weight: normal; text-align: left;"> After practicing many times, Josephine, one of the contestants, has established the probability distribution of her scores, show in the table below. <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">

<span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;"> Crystal, another contestant, has also practiced many times. The probability distribution for her scores is shown in the table below.
 * Josephine's Distribution**
 * <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;"> Score || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">16 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">17 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">18 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">19 ||
 * <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;"> Probability || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;"> 0.10 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">0.30 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">0.40 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">0.20 ||


 * Crystal's Distribution**
 * <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">Score || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">17 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">18 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;"> 19 ||
 * <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">Probability || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">0.45 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">0.40 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">0.15 ||

<span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">(a) Calculate the expected score for each player.

(b) Suppose that Josephine scores 16 and Crystal scores 17. The difference (Josephine minus Crystal) of their scores is -1. List all combinations of possible scores for Josephine and Crystal that will produce a difference (Josephine minus Crystal) of -1, and calculate the probability for each combination.

(c) Find the probability that the difference (Josephine minus Crystal) in their scores is -1. (d) The table below lists all the possible differences in the scores between Josephine and Crystal and some associated probabilities.

Distribution (Josephine minus Crystal)
<span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">**Complete the table and calculate the probability that Crystal’s score will be higher than Josephine’s score.
 * = <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;"> Difference ||= <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">-3 ||= <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">-2 ||= <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">-1 ||= <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">0 ||= <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">1 ||= <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">2 ||
 * = <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;"> Probability ||= <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">0.015 ||= <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;"> 0.00 ||= <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">0.00 ||= <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;"> 0.325 ||= <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">0.260 ||= <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: left;">0.090 ||

Answer!! Click here and scroll down to question number three to see the answer** [] <span style="color: #00ff00; display: block; font-family: 'Comic Sans MS',cursive; font-size: 196.196%; text-align: left;"> GAMES GAMES GAMES FUN FUN FUN <span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 13px; text-align: left;">Here is a website that simulates probabilities with a die. <span style="color: #0000ff; font-family: Tahoma,Geneva,sans-serif;">__http://highered.mcgraw-hill.com/sites/dl/free/0072868244/124727/Probability.html__

<span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 13px; line-height: 19px; text-align: left;">Everything you ever needed!!!!! []

Probability Coin Toss [|http://bcs.whfreeman.com/fapp7e/content/cat_010/probability.html]