Chapter 12- Topics In College Mathematics
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Experiment
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controlled operation that yields a set of results
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Outcomes
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possible results of an experiment
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Event
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subcollection of the outcomes of an experiment
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Empirical Probability
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P(E)- relative frequency of occurrence of an event and is determined by actual observations of an experiment.
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Theoretical Probability
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determined through a study of the possible outcomes that can occur for the given experiment.
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P(E) is read:
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\"P of E\"
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Formula for Empirical Probability
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number of times event E has occurred/ total number of times experiment has been performed ex: 3 out of the 5 batches of cookies were burned. P(E)- 3/5
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The probability of an event, whether empirical or theoretical, is always:
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a number between 0 and 1.
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Probability may be expressed as:
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a decimal number or a fraction.
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Empirical probability is used when:
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probabilities cannot be theoretically calculated.
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Law of Large Numbers:
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probability statements apply in practice to a large number of trials, not to a single trial. It is the relative frequency over the long run that is accurately predictable, not individual events of precise totals.
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Equally likely outcomes:
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If each outcome of an experiment has the same chance of occurring as any other outcome.
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Probability formula (theoretical)
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number of outcomes favorable to E/ total number of possible outcomes.
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The probability of an event that cannot occur is:
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zero.
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The probability of an event that must occur is
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one.
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The sum of the probabilities of all possible outcomes of an experiment is:
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one.
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Odds against an event formula
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event fails to occur/event occurs = failure/success
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Odds against an event is:
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ratio of the probability that the event will fail to occur.
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odds in favor of an event
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ratio of the probability that the event will occur to the probability that the event will fail to occur.
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Expected value (expectation) formula:
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E= p1(a1)+p2(a2)+p3(a3)....pn(an)
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to turn a % into a decimal:
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divide the number by 100. ex: 27%= 0.27
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Expected value:
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the expected gain or loss of an experiment over the long run
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if an individual expects to have a gain in the long run, the expected value is:
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positive
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Fair price formula:
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Expected value+cost to play
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E=
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(win)(winnings)+(lose)(losses)
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Sample space:
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a list of all possible outcomes of an experiment
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Counting principle
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Sample space= M(N)
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With replacement formula:
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*# of ways stays the same, since it can be replaced. Multiply the number of ways together (as many as needed) ex: 2 cards out of 10= card 1 has 10 ways. card 2 has 10 ways 10*10= 100 ways.
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With replacement means:
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whatever is chosen can be put back, giving the next option the same amount of choices.
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without replacement means:
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whatever is chosen is NOT put back, giving the next option one less from the amount of choices.
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without replacement formula:
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ex: 2 cards out of 10 card 1 has 10 ways card 2 has 9 ways 10*9= 90 ways
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counting principle ex with replacement:
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four batteries, three selected at random, with replacement: 4(total batteries)*3(taken out at a time)= 4 to the 3rd power. 4*4*4= 64 sample points
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counting principle ex without replacement
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four batteries, three selected at random, without replacement: 4(total batteries)*3(taken out at a time)= 4*3*2 (3 total batteries)= 24 sample points
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compound probability problems:
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probability problems that contain the words 'and' or 'or'
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independent events
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(for two events A and B) the occurrence of either event in no way affects the probability of the occurrence of the other event.
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Experiments done with replacement result in _____ events.
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independent.
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Addition formula for probability:
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P(A or B)= P(A)+P(B)- P(A and B)
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Conditional probability:
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probability of E2 happening, given that E1 has happened
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When a question states\"what are the chances of\"....... \"given that\".....
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always group FIRST according to the given.
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The symbol for n factorial is:
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n!
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permutations formula:
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nPr= n!/(n-r)!
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(EX) notation that expresses the number of permutations of 6 items taken 2 at a time
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6P2 (total number) P (how many is taken at a time)
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! is:
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the product (multiplying) of all the natural numbers less than the number and itself.
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example of != 4!=
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4*3*2*1= 24
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sample problem of Permutations: 10P4
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10!/10-4!= 10!/6! (this means that you take the last 6 from the problem) 10*9*8*7 (last 6 are gone.) this eliminates the denominator as well. 10*9*8*7= 5040
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6P6 in this equation, it cancels itself out, so just solve as if it were just 6!
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6*5*4*3*2*1= 720
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11 members.. 3 chosen at a time. how many arrangements?
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number*people chosen at a time 11*10*9= 990
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3 letters and 2 digits. with repetition
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26*26*26*10*10 (26 letters) (0-9= 10 numbers)
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3 letters and 2 digits without repetition
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26*25*24*10*9 26 letters, minus one each time. (for 3 letters) 10 numbers, minus one each time for 2 digits.
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3 digit code with digits 1-9 repetition allowed
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9*9*9 9 numbers 3 digits
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combination
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distinct group of objects without regard to their arrangement
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if order of arrangement is important,
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permutations are used
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if order of arrangement is not important,
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combinations are used
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C and P
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combinations and permutations
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Combinations formula
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same as permutations formula, but with added r on denominator
