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SIMULATION THE ACT OF IMITATING AN ACTUAL EVENT, CONDITION, OR SITUATION. WE OFTEN USE SIMULATIONS TO MODEL EVENTS THAT ARE TOO LARGE OR IMPRACTICAL TO LITERALLY TEST. FOR EXAMPLE, IF YOU WANTED TO KNOW THE PROBABILITY OF HAVING A BOY OR GIRL YOU WOULD NOT LITERALLY ATTEMPT THE EVENT. YOU WOULD RUN A SIMULATION. LET’S TRY IT. PROCEDURE 1.Run a simulation to find the gender of each child in 10 families. 2.Each family will have 3 children. 3.Flip a coin to determine the sex of each child. 4.Heads = Male 5.Tails = Female 6.Record your results in the table Family 1 2 3 4 5 6 7 8 9 10 Child 1 Child 2 Child 3 ANALYSIS • How many families had three male children? Female children? • How many of your families had the same order of male and female children? • How many different combinations of offspring are possible in this simulation? • Did anyone else in the class have exactly the same simulation results as you? • Why is the gender of each child an independent event? HOW CAN I DETERMINE IF THE PROBABILITY IS INDEPENDENT OR DEPENDENT? INDEPENDENT DEPENDENT • WHATEVER HAPPENS IN ONE EVENT HAS ABSOLUTELY • WHAT HAPPENS DURING THE SECOND EVENT DEPENDS NOTHING TO DO WITH WHAT WILL HAPPEN NEXT UPON WHAT HAPPENED BEFORE INDEPENDENT EVENTS ARE INDEPENDENT BECAUSE… 1. THE TWO EVENTS HAVE NOTHING TO DO WITH ONE ANOTHER 2. 3. OR YOU REPEAT THE SAME ACTIVITY, BUT YOU REPLACE THE ITEM THAT WAS REMOVED. OR YOU REPEAT AN EVENT WITH AN ITEM WHOSE NUMBERS WILL NOT CHANGE (SPINNERS / DICE / ETC.) TEST YOUR KNOWLEDGE… ARE THE FOLLOWING INDEPENDENT OR DEPENDENT EVENTS? YOU TOSS TWO DICE AND GET A 5 ON BOTH OF THEM INDEPENDENT YOU HAVE A BAG OF MARBLES: 4 WHITE, 5 BLACK, 3 BLUE, 6 PURPLE, AND 10 GREEN. YOU PULL ONE MARBLE OUT OF THE BAG, LOOK AT THE COLOR AND PUT IT BACK IN THE BAG. THEN, YOU CHOOSE ANOTHER MARBLE. INDEPENDENT YOU PULL A QUEEN OF DIAMONDS, THEN A 3 OF SPADES, AND FINALLY A 10 OF HEARTS FROM A DECK OF CARDS WITHOUT PUTTING ANY BACK IN. DEPENDENT The probability of an event is a number between 0 and 1 that indicates the likelihood the event will occur. P=0 Event will not occur P = 1/2 Event is equally likely to occur / not occur You can express a probability as a fraction, a decimal, or a percent. For example: 1 , 0.5, or 50%. 2 P=1 Event is certain to occur THEORETICAL VS. EXPERIMENTAL PROBABILITY THEORETICAL • IN THEORY, THE PROBABILITY OF AN EVENT WILL HAPPEN AS A RESULT OF THE NUMBER OF FAVORABLE OUTCOMES DIVIDED BY THE TOTAL NUMBER OF OUTCOMES P (A) = number of outcomes in A total number of outcomes EXPERIMENTAL • ONE CONDUCTS A PHYSICAL EXPERIMENT TO DETERMINE THE PROBABILITY OF AN EVENT OCCURRING HOW DO YOU FIND THE PROBABILITY? • THE PROBABILITY OF TWO INDEPENDENT EVENTS, A AND B, IS EQUAL TO THE PROBABILITY OF EVENT A TIMES THE PROBABILITY OF EVENT B. • THE PROBABILITY OF TWO DEPENDENT EVENTS, A AND B, IS EQUAL TO THE PROBABILITY OF EVENT A TIMES THE PROBABILITY OF EVENT B. HOWEVER, THE PROBABILITY OF EVENT B NOW DEPENDS ON EVENT A. P(A, B) = P(A) P(B) “AND” INTERSECTIONS (∩) INDEPENDENT • Remember, this means the occurrence of one does not change the probability of the other occurring. • 𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃 𝐴 ∙ 𝑃(𝐵) DEPENDENT • Remember, this means the occurrence of one affects the probability of the other occurring. •𝑃𝐵𝐴 means the probability of B given A has already occurred • 𝑃 𝐴 𝑎𝑛𝑑 𝐵 =𝑃 𝐴 ∙𝑃 𝐵 𝐴 “AND” INTERSECTIONS (∩) INDEPENDENT EVENT Example: Suppose you spin each of these spinners. What is the probability of spinning a star and a “B”? 3 P(star) = (3 stars out of 8 outcomes) 8 1 P(B) = (2 “B”s out of 6 outcomes) 3 3 1 3 1 ∙ = = P(star, B) = 8 3 24 8 Slide 13 DEPENDENT EVENT Example: There are 6 black socks and 8 white socks in your dresser drawer. If you get dressed in the dark and take a sock without looking and then take another sock without replacing the first, what is the probability that you will get 2 black socks? P(black first) = 6 3 or 14 7 5 P(black second) = (There are 13 socks left and 5 are black) 13 Therefore… P(black, black) = 3 5 15 or 7 13 91 “OR” UNIONS (U) A B MUTUALLY EXCLUSIVE EVENTS • TWO EVENTS ARE MUTUALLY EXCLUSIVE IF THEY CANNOT OCCUR AT THE SAME TIME. • DISJOINT IS ANOTHER WORD THAT MEANS MUTUALLY EXCLUSIVE 𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡: 𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 0 • IF TWO EVENTS ARE MUTUALLY EXCLUSIVE, THEN THE PROBABILITY OF EITHER OCCURRING IS THE SUM OF THE PROBABILITIES OF EACH OCCURRING 𝑃 𝐴 𝑜𝑟 𝐵 = 𝑃 𝐴 + 𝑃(𝐵) NON-MUTUALLY EXCLUSIVE EVENTS • IN EVENTS WHICH AREN’T MUTUALLY EXCLUSIVE, THERE IS SOME OVERLAP. • WHEN P(A) AND P(B) ARE ADDED, THE PROBABILITY OF THE INTERSECTION (AND) IS ADDED TWICE. • TO COMPENSATE FOR THAT DOUBLE ADDITION, THE INTERSECTION NEEDS TO BE SUBTRACTED. 𝑃 𝐴 𝑜𝑟 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 𝑎𝑛𝑑 𝐵) COMMON EXAMPLES: • WHAT ROUTE TO TAKE TO WORK TO AVOID DELAYS OR ACCIDENTS • HOW TO DRESS FOR THE WEATHER (HOT/COLD, RAIN/SNOW, ETC.) • GAMBLING GAMES OF ALL KINDS • WHERE TO EAT TO AVOID FOOD POISONING CAREER EXAMPLES: • MEDICAL CAREERS • ACTUARIAL SCIENCES • BIO-MATHEMATICIANS • BIOMEDICAL SCIENCES • ENVIRONMENTAL AND ECOLOGICAL SCIENCES • FINANCES • ENGINEERING • NURSING • PHARMACEUTICALS • PUBLIC POLICY • QUALITY IMPROVEMENT • GOVERNMENT SERVICES • RISK ANALYSIS • SURVEY RESEARCHERS • MARKETING • SO MUCH MORE!!!