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Probability

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					Probability:


  1. Permutations: used to determine the number of arrangements on n
     objects chosen r at a time when the order is important.


                                n!
     Notation used:   nP 
                        r
                             n  r !

  2. Factorial: used after a number to indicate the product of all numbers
     equal to that number and below it.
     Ex. 5! = 5(4)(3)(2)1= 120

      Notation used: n! = n(n-1)(n-2)…….2(1)
          Where 0!=1 and 1!=1




  The fundamental counting principle: If event A occurs in r number of
ways and event B occurs in s number of ways, then the number of ways that
both events can occur is r  s .
     Ex. Joan has 3 shirts and 5 pairs of pants in her closet. How many outfits
can she make with these clothes?
                 3 5  15
     ex. Estabon wants to make a sandwich. He has 3 types of bread to
choose from, 2 types of cheese, 4 types of meats, and either mustard or mayo.
How many different sandwiches can he make?
                 3 2  4  2  48
  Back to permutations:
     Ex.   7   P3 


      Ex. The chess club decides to elect a slate of officers. First they will elect
a president, then a vice president, and finally a treasurer. If there are 10
members in the club, how many different groups of officers are possible?
      (the order in which these officers are elected is important because each
elected office holds a different position.)

               10   P3 


    ex. How many 3-letter arrangements can be made using the letters in the
work SHELF?

               5   P3 


    ex. How many 3-letter arrangements can be made using the letters in the
work EAGLE?
                   P3
               5
                      
                   2!
   Combinations: used to determine the number of possible arrangements of
n objects chosen r at a time, when the order does not matter.
                                   n!
     Notation used:   n Cr 
                               r!n  r !



       Ex. A group consisting of 40 members wants to select a committee of 4
of its members from their group of 40 people.




     Ex. The astronomy club has 25 members of 11 girls and 14 boys. The club
wants to form a committee of 4 members to organize their annual Family
Astronomy night.
     a. How many committees can be formed consisting of any of the 25
        members?
     b. How many committees can be formed consisting of only girls?
     c. How many committees can be formed consisting of 2 girls and 2 boys?
     d. What is the probability that a committee formed consists of 2 girls
        and 2 boys from the club?
Probability of “n” things happening EXACTLY “r” times:
      n   Cr ( p)r (q)nr

     where n is the total number of trials
                    r is the number of successes
                    p is the probability of success
                    q is the probability of failure ( 1 – p )


ex. The probability of it raining is 3/5. What is the probability of it raining 4
days next week?




Ex. Given a fair die, what is the probability of rolling a 2 on 3 of the next 4
rolls of the die?
Probability of “at most” r successes out of n trials:
       This means the most successes that will occur is “r” successes.
       If we roll a die 6 times, what is the probability that we roll a 4 AT MOST
            3 times. This means we can roll a 4 no times (or 0 times), or 1 time
or
            2 times or 3 times. But no more than 3 times because of our initial
            statement of “at most” 3 times!!
       We have to set up a probability for each of the possible outcomes and
then
       Add them together.




       If we toss a coin 4 times, what is the probability that we get at most two
            Tails?
       We can get no tails, one tail or two tails, but no more than two!!
Probability of “AT LEAST” r successes out of n trials:
       This means we can get r successes or more than r successes, but no more
than
       N successes. (because we only have n trials)


Ex. Given a spinner, divided equally into 5 sectors, numbered 1-5, what is the
probability of getting an odd number AT LEAST 2 times on 4 spins of the
spinner?
       We can get an odd number 2 times, 3 times or 4 times, but no more than
4
       Times.
       We have to set up a probability for each of the outcomes and then add
them
       Together.
Binomial Expansion: This is when we expand any binomial raised to any
power.
     The higher the power, the harder and more work it is for us! So we have
a
     Formula we use to do this. It is on your reference sheet. It sort of
reminds of
     The probability formula.


Ex. Expand the following:
     1. 2x  y 
                3




     2. cos  3
                        4




                    5
        1     
     3.   4a 
        2     




Sometimes, we may only be asked to find a specific term or the coefficient of a
specific term. If that is the case, we need only find that term and not expand
the whole binomial. If we need to find the coefficient, then just take the
number in front of the variable. Also, when asked to find the constant term,
this is the term without a variable attached to it.


Examples from worksheet:

				
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posted:8/28/2012
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