1 Chapters 6-8

UNIT 2 VOCABULARY – Chap 6 2

( 2) THE NOTATION “P” REPRESENTS THE TRUE PROBABILITY OF AN EVENT HAPPENING, ACCORDING TO AN IDEAL DISTRIBUTION. * THIS IS A PARAMETER: A SET, FIXED, CONSTANT VALUE WHICH IS SOMETIMES KNOWN AND SOMETIMES UNKNOWN, DEPENDING ON HOW COMPLICATED THE SPECIFIC PROBLEM IS. * USUALLY, WE DON’T KNOW THE VALUE OF P, SO WE SOMETIMES TRY TO ESTIMATE IT WITH…… 3

“P-HAT”, WHICH REPRESENTS AN EXPERIMENTAL PROBABILITY, OR ONE THAT COMES FROM A SET OF DATA INSTEAD OF AN IDEALIZED FORMULA. * THIS IS A STATISTIC: A VARIABLE, NON-CONSTANT VALUE WHICH CHANGES EVERY TIME THE EXPERIMENT IS RUN OR THE DATA IS COLLECTED. * SINCE P-HAT TAKES ON A SET OF VALUES, THESE CAN BE GRAPHED, JUST LIKE OTHER SETS OF NUMBERS. * WE USUALLY RUN AN EXPERIMENT OR COLLECT A SET OF DATA SO THAT, WHEN WE CALCULATE THE VALUE OF THE STATISTIC, WE CAN USE THIS VALUE TO ESTIMATE THE TRUE VALUE OF THE ASSOCIATED PARAMETER. 4

(3) The Sample Space of an event is the set of all outcomes as measured by the definition of the variable we are assessing. For example, if I flip a coin and am interested in recording which side is UP, then the sample space would be {heads, tails}. On the other hand, if I flipped three coins and were interested in how many heads I get – recording this value as X – then the sample space would be {0,1,2,3}. If I were interested in whether or not it would snow today, then the sample space would be {snows, doesn’t snow}. 5

(3) Sample Space, continued…… Not everything in the Sample Space has to have the same probability of happening. However, the sum of all of the probabilities of the events in the sample space must add up to 1 (100%), since this is everything which is possible to happen. 6

(4) Probability Distribution > a table with all of the values that X can take (its sample space) and the probabilities of each of those values > the sum of all of the probabilities that are listed in a probability distribution should equal 100% 7

(5) Mutually Exclusive Events can’t happen at the same time. > “Disjoint” is another term for Mutually Exclusive > For example, if I roll a die, then the events “I get a multiple of 3” and “I get a multiple of 4” are mutually exclusive. 8

(5) (continued) P (A or B) = P(A) + P(B) – P(A and B at the same time) Note: If A and B are mutually exclusive, then they can’t happen at the same time. Thus, P(A and B at the same time) would equal 0, right? So the formula would edit down to this: P (A or B) = P(A) + P(B) 9

(6) If I tell you ahead of time that I am somehow restricting the domain, then this will be a Conditional Probability. > Words that might alert you: “if”, “given”, “when” Tricky formula: Try not to use this formula all of the time; just use common sense! 10

(7) Two events are called complements of each other if…… * They are “opposite” of each other, and * Between the two of them, they make up the entire sample space, and * Their probabilities add up to 100%. 11

(7) (continued) For example, if I were to select a student at random from this class, then “I pick a boy” and “I pick a girl” would be complementary. Another example: if Ryan was shooting five free throws, then “he misses them all” and “he makes at least one shot” would be complements. 12

(8) INDEPENDENCE  Informally, this refers to the outcome of one event not having an influence on the outcome of another event.  Here it is, in more technical language: “Two events – A and B – are said to be independent if knowing that one occurs does not change the probability that the other occurs.” 13

(8) More Independence:  There are two, equivalent formulas for determining independence. The one you use depends on the individual problem you are doing: 14

(i) A and B are independent if and only if P(A and B) = P(A) * P(B) (book pg 351) (ii) A and B are independent if and only if P(B given A) = P(B)(book pg 375) 15