11.3 Permutations

• In a permutation, the order of the items is important.
• In a combination the order is not important.
• _nCr = n!/[r!(n-r)!]
• Order is the important question.
• Look at powerball results:
  • They are always given in order.
  • But they are drawn from balls from a hopper in any order.
  • So in this case order does not matter.
  • How many different power ball combinations are there?

    55C5 * 42C1 = 147,107,962
           

• How about the Pa lottery game Mix and Match?
  • "Get a mix & match play slip at any Pennsylvania Lottery numbers games retailer. Pick five numbers between 1 and 19 in the positions you think they will be drawn by the Pennsylvania Lottery. Or, you can have the computer randomly choose your numbers through Quick Pick. Each play costs $2." (from www.palottery.state.pa.us)
  • Combination or permutation?

    19P5 = 1,395,360
           

  • How would this game change if order were not important?

    19C5 = 11,682 
           

• Do examples

  1-4 all
  
  Any 5 - 28 evens you want.
  
  30, 32, 
  
  Classify 38-46, do if we want.
  

11.4 Fundamentals of Probability, Theoretical Probability

• Probability is a branch of mathematics concerned with how likely it is something will happen.
• We are never told if an event will happen or not, but just how likely it is.
• An experiment is some form of action on which we are computing the probability.
• An event is one possible outcome of an experiment.
• The sample space of an event is a list of all possible outcomes or events.
• The probability of an event E, P(E) is n(E)/n(S)
• Example:

  If we toss a six sided die, what is the probability that 
    E1 a 1 shows:  
           n(E1) = 1 
  	 n(S) = 6
  	 P(E1) = 1/6
    E2 an even number shows:  
           n(E2) = 3 
  	 n(S) = 6
  	 P(E2) = 3/6  = 1/2
    E3 a number less than 5 shows:  
           n(E3) = 4 
  	 n(S) = 6
  	 P(E3) = 4/6  = 2/3
    E4 a number greater than 6 shows:  
           n(E4) = 0 
  	 n(S) = 6
  	 P(E4) = 0/6
    E5 a number less than or equal to  6 shows:  
           n(E4) = 6 
  	 n(S) = 6
  	 P(E4) = 6/6 = 1
     

• Notice we can't have a probability less than 0 or greater than 1.
• A Deck of cards
  • Numbers: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King
  • Hearts, Diamonds, Spades, Clubs
  • Red, Black
  • Face cards = Jack, Queen, King
• Example:

  Experiment: A single card is drawn from a deck.
  E1 the card is a 7
     n(E1) = 4
     n(S) = 52
     P(E1) = 5/52 = 1/13
  E2 the card is a 4 of diamonds
     n(E2) = 1
     n(S) = 52
     P(E2) = 1/52
  E3 the card red 
     n(E3) = 26
     n(S) = 52
     P(E3) = 26/52 = 1/2
  E4 the card red 
     n(E4) = 26
     n(S) = 52
     P(E4) = 26/52 = 1/2
     

• Look at the problems

  28, 30, 36, 38, 42, 44, 46