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Chapter 4 – Probability and Counting Rules section 4.3 – Multiplication Rules and Conditional Probability. Multiplication Rules for Probability. Independent Events Two events are independent if the fact that A occurs does not affect the probability of B occuring. Flip a coin 5 times…

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Chapter 4 – Probability and Counting Rulessection 4.3 – Multiplication Rules and Conditional Probability

multiplication rules for probability
Multiplication Rules for Probability

Independent Events

Two events are independent if the fact that A occurs does not affect the probability of B occuring.

  • Flip a coin 5 times…
  • drawing a card, replacing it, drawing again
multiplication rules for probability1
Multiplication Rules for Probability

Multiplication Rule 1

When two events are independent, the probability of both occuring is

P(A and B) = P(A)P(B)

  • A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die.
  • A card is drawn and replaced. Find the probability of drawing a queen and then an ace.
  • The multiplication rule can be extended to many events…
multiplication rules for probability2
Multiplication Rules for Probability

Multiplication Rule 2

when two events are dependent, the probability of both occurring is

P(A and B) = P(A)·P(B|A)

  • P(B|A) is a conditional probability
  • It means the probability that B occurs given that A has already occurred.
multiplication rules for probability3
Multiplication Rules for Probability

examples

  • A person owns 30 CD’s. 5 of them are jazz. If 2 CD’s are selected at random, what is the probability that both are jazz music?
    • Are the two selections independent?
  • P(J1and J2) =
  • 53% of residents of a city have homeowner’s insurance. Of these 27% also have car insurance. Find P(home ins. and car ins.).
multiplication rules for probability4
Multiplication Rules for Probability
  • try p 220 # 1 in your notes
multiplication rules for probability5
Multiplication Rules for Probability

Conditional Probability

The probability that the second event B occurs given that the first event A has occurred is found by:

P(B|A) = P(A and B)

P(A)

example
Example
  • P(B and W) is 15/56 and P(B) on the first draw is 3/8. Find the probability of selecting the white chip on the second draw, given that the first chip selected was a black chip.
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The probability that Sam parks in a no-parking zone and gets a ticket is 0.06 and the probability that Sam cannot find a parking space and has to park in the no-parking zone is 0.20. On Tuesday, Sam arrives at school and has to park in a no-parking zone. Find the probability that he will get a ticket.

at least
“At- least”
  • A game is played by drawing 4 cards from an ordinary deck and replacing each card after it is drawn. Find the probability that at least 1 ace is drawn.
    • It is easier to find the probability that no aces are drawn and subtract that answer from 1.
multiplication rules for probability6
Multiplication Rules for Probability

turn to page 221

  • try number 11 in your notes
  • try number 13
practice
Practice
  • In class
    • p 220 (2-30 even)