Unit 1: Probability. Experimental Probability: - probability that came from a simulation such as tossing dice, coins etc.

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1 pplied Math 0 Unit : Probability Unit : Probability.: Experimental and Theoretical Probability Experimental Probability: - probability that came from a simulation such as tossing dice, coins etc. inomial Distribution: - a sample space where there is only two outcomes (favorable, non-favorable). randin (Random inomial): generates and displays a random integer from a specified binomial distribution. randin (number of trials, Theoretical Probability of favorable outcome, number of simulations) lways set to To access randin:. Press MTH 2. Use to access PR. Select Option 7 Theoretical Probability: - probability from calculations by using sample space or other formulas. Example : Find the probability of obtaining tails when a coin is tossed 00 times theoretically and experimentally. Theoretical Probability Sample Space of Tossing a Coin = (Head), (Tail) P (Tail) = 2 Experimental Probability using Graphing Calculator. randin (, ½, 00) 2. Summing all trails of favorable outcome. sum (ns) 2nd 2nd LIST STT NS ( ) Experimental P (Tail) = = 0. 6 = (Your answer might be different depending on when you last RESET your calculator.) Copyrighted by Gabriel Tang.Ed.,.Sc. Page.

2 Unit : Probability pplied Math 0 Example 2: Find the theoretical and experimental probability of NOT landing on a using the spinner below if it was spun 00 times Theoretical Probability P () = P 9 ( ) = P ( ) P (NOT ) 9 = Experimental Probability using Graphing Calculator. randin (, 8/9, 00) 2. Summing all simulations of favorable outcome Experimental P ( ) = = Example : What is the theoretical probability of the following from a standard deck of 2 cards? a. ny ace. b. ny number cards. (There are ces in a deck of cards.) P (ce) = 2 There are 6 Number cards in a deck. ({2,,,,6,7,8,9,0} suits) 6 P (Number Cards) = 2 P (ce) = P (Number Cards) = 9 Page 2. Copyrighted by Gabriel Tang.Ed.,.Sc.

3 pplied Math 0 Unit : Probability Example : Find the number of ways to reach the other side of the checkerboard if you can only move forward diagonally from. C D C D Total Pathways = Total Number of Pathways = 89 Example : Determine the number of ways a pinball will pass through set up below. What is the probability that it will exit at? all all Total Pathways = Total Number of Pathways = 8 P (exit ) = 8 2 C D C D Copyrighted by Gabriel Tang.Ed.,.Sc. Page.

4 Unit : Probability pplied Math 0 Example 6: Determine the number of ways you can reach from if the path taken must be in either north or east direction. N Total Number of Pathways = 26 a b. Detour Total Number of Pathways = Page. Copyrighted by Gabriel Tang.Ed.,.Sc.

5 pplied Math 0 c. Unit : Probability Total Number of Pathways = ssignment pg. 0 2 # to 0 Copyrighted by Gabriel Tang.Ed.,.Sc. Page

6 Unit : Probability pplied Math 0-2: Generating and Using Sample Space Sample Space: - a list of all possible outcomes of an experiment. Complement: - probability of LL Non-Favourable outcomes. P () = Probability of event happening. P ( ) = Probability that event will NOT happen. In general, P ( ) P( ) = ( ) or P = P( ) Example : Find the theoretical probability of rolling a using a six-sided dice, and its compliment. Sample Space of a dice = {, 2,,,, 6} P () = 6 ( favourable outcome out of a total of 6 outcomes) P ( ) = P () = P 6 ( ) = 6 Tree Diagram: - using branches to list all outcomes. - following branches one path at a time to list all outcomes. - its limitation lies in the fact that it can only handle individual events having SMLL number of outcomes. Example 2: List the sample space of a family of children. Let event be at most having 2 girls, find P () P. and ( ) st child 2 nd child rd child Outcomes G G G G G G G (,, ) (,, G) (, G, ) (, G, G) (G,, ) (G,, G) (G, G, ) (G, G, G) P () = Probability of at most 2 girls (no girls, girl, and 2 girls) P () = 8 7 (7 favourable outcomes out of a total of 8 outcomes) OR we can say P () = P (all girls) 7 P () = P () = P ( ) = P () = P 8 ( ) = 8 Page 6. Copyrighted by Gabriel Tang.Ed.,.Sc.

7 pplied Math 0 Unit : Probability Sample Space Table: useful when there are TWO items of MNY outcomes for each trial. Example : List the sample space for the sums when 2 fair dice are thrown a. Find the probability of rolling a sum of 8. b. What is the probability of rolling at least an 8? First Dice Second Dice Sum a. P (8) = 6 ( favourable outcomes out of a total of 6 outcomes) b. P (at least a sum of 8) = P (sum of 8 and higher) P (at least a sum of 8) = 6 P (at least a sum of 8) = 2 Example : Without draw a tree diagram, determine the number outcomes will be in the sample space when coins are tossed. What is the probability of tossing at least one head? From Example 2, we noticed that when there were events ( children) of 2 outcomes (boy or girl) each, there were a total of = 8 outcomes. Therefore, it can be assumed that when there are events (tossing four coins) of 2 outcomes (head or tail) each, there are a total of = 6 outcomes. P (at least one head) = P ( or 2 or or heads) P (at least one head) = P (no head) P (at least one head) = P (all tails) P (at least one head) = 6 (There is only outcome out of a total of 6 outcomes to get all tails.) P (at least one head) = 6-2 ssignment: pg. 7 9 # to 0 Copyrighted by Gabriel Tang.Ed.,.Sc. Page 7.

8 Unit : Probability pplied Math 0 -: The Fundamental Counting Principle Counting Principle: - by multiplying the number of ways in each category of any particular outcome, we can find the total number if arrangements. Example : How many outfits can you have if you have different shirts, 2 pairs of pants, pairs of socks, and pair of shoes? There are categories: shirt, pants, socks and shoes. 2 = 2 outfits shirts pants socks shoes When the outcome involves many restrictions, make sure to deal with the category that is the most restrictive first. Example 2: How many -digits numbers are there if a. there is no restriction? b. they have to be divisible by? st digit 2 nd digit rd digit th digit th digit ( to 9) (0 to 9) (0 to 9) (0 to 9) (0 to 9) st digit 2 nd digit rd digit th digit th digit ( to 9) (0 to 9) (0 to 9) (0 to 9) (0 or ) numbers 8000 numbers restriction c. no digits are repeated? d. they have to be less than 0000 with non-repeated digits? st digit 2 nd digit rd digit th digit th digit st digit 2 nd digit rd digit th digit th digit ( to 9) (0 to 9) (0 to 9) (0 to 9) (0 to 9) digit 2 digits digits digits ( to ) (0 to 9) (0 to 9) (0 to 9) (0 to 9) digit 2 digits digits digits 2726 numbers 9072 numbers e. at least two digits that are the same? t least two digits the same = same 2 digits same digits same digits same digits OR t least two digits the same = Total number of ways with no restriction No digits the same = numbers Page 8. Copyrighted by Gabriel Tang.Ed.,.Sc.

9 pplied Math 0 Unit : Probability f. Find the probability of getting a -digits numbers if at least 2 digits have to be the same. P (at least 2 same digits out of a -digits number) = 6 62 Number of ways at least same 2 digits = Total Number of ways or Example : In an pplied Math 0 Diploma Exam, there are multiple-choice questions. Each question contains choices. How many different combinations are there to complete the test, assuming all questions are answered? What is the probability that a student would guess all the answers correctly? For each question, there are ways (choices) to answer. For questions, there are There are factors of fours. Total number of ways to answer the test = or ways P (guessing all the right answers) = = (There is a way better chance to win the lottery compared to guessing all the answers correctly!) Example : Find the number of ways to select 2 individuals from a group of 6 people. Let s suppose the names of the people are,, C, D, E and F. The sample space table would be Order is NOT important! These C D E F repeated combinations do not need to be counted. C C C D E F D D CD DC EC FC Total Number of ways = E E CE DE ED FD F F CF DF EF FE Summary of the Counting Principle YES Restrictions? YES Count the outcomes of the most restricted category first. ORDER Important? NO Multiply the number of ways in each category. - ssignment: pg. 2 2 # to 2 NO.Count Sample Spaces by using tables or tree diagram. 2.Cross out any duplicated results. Copyrighted by Gabriel Tang.Ed.,.Sc. Page 9.

10 Unit : Probability pplied Math 0 -: Independent and Dependent Events Independent Events: - when the outcome of one event does NOT affect the outcomes of the events follow. P ( and ) = P () P () Events and Example : What is the probability of rolling at most a from a dice and selecting a diamond out of a standard deck of cards? P (at most & diamond) = P (at most & diamond) = 6 2 Reduce before multiply 2 2 = 2 P (at most & diamond) = 6 Example 2: Find the probability of getting at least one number correct out a -numbers combination lock with markings from 0 to 9 inclusive. ll numbers are Incorrect (No number is correct) is the Compliment Event of t Least ONE number is correct P (ll numbers are Incorrect) = = (There are 60 numbers from 0 to 9 inclusive.) P (t Least ONE number is correct) = P (ll numbers are Incorrect) 9 P (t Least ONE number is correct) = 60 P (t Least ONE number is correct) % Example : What is the probability of drawing two aces if the first card is replaced (put back into the deck) before the second card is drawn? P (drawing 2 aces) = P (drawing 2 aces) = 2 2 P (drawing 2 aces) = % 69 Page 0. Copyrighted by Gabriel Tang.Ed.,.Sc.

11 pplied Math 0 Unit : Probability Dependent Events: - when the outcome of the first event FFECTS the outcome(s) of subsequent event(s). Example : What is the probability of drawing two hearts from a standard deck of 2 cards? If the question doe NOT say, assume the card is NOT Replaced! 2 One less heart remains in the deck after the st draw. P (two hearts) = 2 One less card total remains in the deck after the st draw. P (two hearts) = 7 P (two hearts) = % Example : bout 0% of the population likes sci-fi movies, while 80% of the population likes comedies. If % of the population likes both sci-fi and comedies, determine whether preferences for sci-fi and comedies are independent events. Justify your answer mathematically. P (sci-fi) = 0% = 0.0 P (comedies) = 80% = 0.80 If sci-fi and comedies are independent events, then P (sci-fi & comedies) = P (sci-fi) P (comedies) P (sci-fi & comedies) = P (sci-fi & comedies) = 0.2 (independent events) Since the question indicates P (sci-fi & comedies) is % = 0., they should be considered as DEPENDENT EVENTS. Example 6: There are two identical looking containers. Each contains different number of black and white balls of the same size. Container has 8 balls in total and of those are white. Container has 0 balls in total and of those are black. If a person has a choice to pick one ball out of these two containers, what is the probability of selecting a black ball? P (black) = P (Container & lack) OR P (Container & lack) OR means DD Container Container P (black) = P () P (lack) P () P (lack) P (black) = P (black) = 6 20 P (black) = = 0.2 =.2% 80 - ssignment: pg. # to 9, Copyrighted by Gabriel Tang.Ed.,.Sc. Page.

12 Unit : Probability pplied Math 0 -: Mutually Exclusive Events Mutually Exclusive Events: - when events & CNNOT occur at the SME TIME. - either event occurs OR event occurs. Mutually Exclusive Events P ( or ) = P () P () NO Overlap Example : card is drawn from a standard deck of 2 cards. What is the probability that it is a red card or a spade? Since there is no card that is OTH a spade and red, the events are mutually exclusive. 26 P (red) = = P (spade) = = P (red or spade) = P (red) P (spade) P (red or spade) = 2 P (red or spade) = Example 2: Two dice are rolled. What is the probability that either the sum is or the sum is 8? First Dice Second Dice Sum Since you can either roll a sum of or a sum of 8, but NOT OTH, the events are mutually exclusive. P () = 6 2 P (8) = 6 P ( or 8) = P () P (8) 2 P ( or 8) = P ( or 8) = 6 Page 2. Copyrighted by Gabriel Tang.Ed.,.Sc.

13 pplied Math 0 Unit : Probability Non-Mutually Exclusive Events: - when events & CN occur at the SME TIME. - the probability of the overlapping area, P ( and ), is subtracted. Non-Mutually Exclusive Events P ( or ) = P () P () P ( and ) Overlapping rea P ( and ) = P () P () (if and are independent events) Example : Two dice are rolled. What is the probability that one of the dice is or the sum is 8? First Dice Second Dice Sum Since you can have sums of 8 that have on one dice, these events are non-mutually exclusive. 0 P (one of the dice is ) = 6 (sum of 6 has two s Not included) 2 P (8) = P ( on one dice and sum is 8) = 6 6 P ( on one dice or sum of 8) = P () P (8) P ( on one dice & sum of 8) 0 2 P ( on one dice or sum of 8) = P ( on one dice or sum of 8) = 6 Example : Out of 200 people, 80 of them like sci-fi and 60 like comedies. 70 of them like both sci-fi and comedies. Find the probability that someone will like either sci-fi or comedies. Since you can have people prefer OTH sci-fi and comedies, these events are non-mutually exclusive P (sci-fi) = P (comedies) = P (sci-fi and comedies) = P (sci-fi or comedies) = P (sci-fi) P (comedies) P (sci-fi and comedies) P (sci-fi or comedies) = P (sci-fi or comedies) = = = 0.8 = 8% Copyrighted by Gabriel Tang.Ed.,.Sc. Page.

14 Unit : Probability pplied Math 0 Example : In our society, the probability of someone suffering from heart disease is 0.6 and the probability of developing cancer is 0.2. If these events are independent and non-mutually exclusive, what is the probability that you will suffer neither illness? P (Heart Disease) =P (H) = 0.6 P (Cancer) = P (C) = 0.2 P (H and C) = P (H) P (C) = (independent events) P (H and C) = 0. P (H or C) = P (H) P (C) P (H and C) P (H or C) = (non-mutually exclusive events) P (H or C) = H C P (No Heart Disease NOR Cancer) = P ( H and C ) rea OUTSIDE the circles P ( H and C ) = P (H or C) P ( H and C ) = P ( H and C ) = 0. =.% Example 6: Two cards are drawn from a standard deck of 2 cards. What is the probability that both cards are face cards or both are hearts? The question did not mention replacement. Therefore, we can assume there is NO replacement (Dependent Events). Since you can have OTH face cards and hearts, these events are non-mutually exclusive. 2 2 P (face cards) = = P (hearts) = = P (face cards and hearts) = = (K, Q, and J of Hearts) 2 2 P (face cards or hearts) = P (face cards) P (hearts) P (face cards and hearts) P (face cards or hearts) = P (face cards or hearts) = = 0.06 = 0.6% 2 - ssignment: pg. 7 9 # to Page. Copyrighted by Gabriel Tang.Ed.,.Sc.