Conditional Probability 12.7

• The probability of event E₂ occurring, given that event E₁ has happened is called Conditional Probability and is written P(E₂ | E₁) .
• P(E₂ | E₁) is read "the probability of E₂ given E₁.
• It means that if E₁ has happened, what are the chances of E₂ happening?
• P(E₂ | E₁) = n(E₁ and E₂) / n(E₁)
• The formula means that we have eliminated some of the search space.
• A card is drawn at random, if the card is red, what is the probability that it is a diamond?
  • n(Red and Diamond) = 13
  • n(red) = 26
  • P(Diamond given it is Red) = 13/26
• Problem 6, page 787
• Problem 18
• Problem 42

Combinations 12.9

• A combination is a group of objects without respect to order.
• Ways to order two out of 4 dice (rgbp)

        If order matters     If order does NOT matter
           rg rb rp              rg rb rp
           gr gb gp              gb gp   (gr is the same as rg)
           br bg bp              bp    (bg=gb and br=rb)
           pr pg pb              (all have been used)
      

• _nC_r = n!/[(n-r)!r!]
• ₄C₂ = 4!/(2!)(2!) = 4×3/2 = 6
• ₄P₂ = 4!/(2!) = 4×3 = 12
• The most important question here, "Does the order of the items matter?"
• Look at the problems 22-44, does the order matter?

Homework

• Page 787: 5 to 55 odd.
• Page 807: 21 to 41 odd.