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PROBABILITY

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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.

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PROCEDURE FamilyChild 1Child 2Child 3 1 2 3 4 5 6 7 8 9 10 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

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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?

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HOW CAN I DETERMINE IF THE PROBABILITY IS INDEPENDENT OR DEPENDENT? INDEPENDENT WHAT HAPPENS DURING THE SECOND EVENT DEPENDS UPON WHAT HAPPENED BEFORE WHATEVER HAPPENS IN ONE EVENT HAS ABSOLUTELY NOTHING TO DO WITH WHAT WILL HAPPEN NEXT DEPENDENT

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INDEPENDENT EVENTS ARE INDEPENDENT BECAUSE… 1. THE TWO EVENTS HAVE NOTHING TO DO WITH ONE ANOTHER OR 2. YOU REPEAT THE SAME ACTIVITY, BUT YOU REPLACE THE ITEM THAT WAS REMOVED. OR 3. YOU REPEAT AN EVENT WITH AN ITEM WHOSE NUMBERS WILL NOT CHANGE (SPINNERS / DICE / ETC.)

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TEST YOUR KNOWLEDGE… ARE THE FOLLOWING INDEPENDENT OR DEPENDENT EVENTS? YOU TOSS TWO DICE AND GET A 5 ON BOTH OF THEM 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. 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. INDEPENDENT DEPENDENT

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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 P = 1 Event is certain to occur You can express a probability as a fraction, a decimal, or a percent. For example:, 0.5, or 50%. 1 2

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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 EXPERIMENTAL ONE CONDUCTS A PHYSICAL EXPERIMENT TO DETERMINE THE PROBABILITY OF AN EVENT OCCURRING THEORETICAL VS. EXPERIMENTAL PROBABILITY P (A) = total number of outcomes number of outcomes in A

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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.

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“AND” INTERSECTIONS ( ∩ ) INDEPENDENTDEPENDENT

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“AND” INTERSECTIONS ( ∩ )

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Example: Suppose you spin each of these spinners. What is the probability of spinning a star and a “B”? P(star) = (3 stars out of 8 outcomes) (2 “B”s out of 6 outcomes) P(B) = P(star, B) = Slide 13 INDEPENDENT EVENT

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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 second) = (There are 13 socks left and 5 are black) P(black first) = Therefore… P(black, black) =

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“OR” UNIONS (U) AB

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MUTUALLY EXCLUSIVE EVENTS TWO EVENTS ARE MUTUALLY EXCLUSIVE IF THEY CANNOT OCCUR AT THE SAME TIME. DISJOINT IS ANOTHER WORD THAT MEANS MUTUALLY EXCLUSIVE IF TWO EVENTS ARE MUTUALLY EXCLUSIVE, THEN THE PROBABILITY OF EITHER OCCURRING IS THE SUM OF THE PROBABILITIES OF EACH OCCURRING

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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.

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LET’S APPLY THIS TO THE WORLD… WHAT ARE SOME SITUATIONS IN WHICH YOU MAY NEED TO USE PROBABILITY TO MAKE A DECISION? WHAT CAREERS CAN YOU THINK OF THAT WOULD INCORPORATE PROBABILITY/STATISTICS IN DECISION MAKING?

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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

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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!!!

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LET’S LOOK AT SOME REAL SITUATIONS TOGETHER…

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