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Aim: What is the multiplication rule?

Aim: What is the multiplication rule?. Independent Event. Independent Event: the probability of an event occurring is not affected by the previous event Example: flipping a coin, having a boy/girl. What is the multiplication rule?.

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Aim: What is the multiplication rule?

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  1. Aim: What is the multiplication rule?

  2. Independent Event • Independent Event: the probability of an event occurring is not affected by the previous event • Example: flipping a coin, having a boy/girl

  3. What is the multiplication rule? • Multiplication Rules can be used to find the probability of two or more events that occur in a sequence • Example: if a coin is tossed and then a die is rolled, one can find the probability of getting a head on he coin and a 4 on the die • These two events are independent since the outcome of the first event does not affect the probability outcome of the second event!

  4. Multiplication Rule 1 • When two events are independent, the probability of both occurring is • P(A and B) = P(A) * P(B) • Example: 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. P(head and 4) = P(head) * P(4)

  5. Extending to three events… • Multiplication Rule 1 can be extended to 3 or more events: • P(A and B and C and …) = P(A) * P(B) * P(C) * …

  6. The “fifth” rule of probability! • There is a ‘fifth’ rule for independent pairs of events • Rule 5: Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent • P(A and B) = P(A)×P(B) • Sometimes called the multiplication rule for independent events.

  7. Example • Since the 4 outcomes above are equally likely, then the probability of the second flip being heads is not affected by the result of the first flip. • In each case, the probability of heads is still ½ (doesn’t matter which column you are in, there are still two outcomes with one satisfying the condition of 2nd flip being heads.

  8. Dependent Events • When the occurrence of the first event changes the probability of the occurrence of the second event • Example: suppose a card is drawn from a deck and it is not replaced and then a second card is drawn

  9. Finding the probability of Dependent Events • To find the probability of dependent events, use the multiplication rule with a modification in notation…known as conditional probability • The notation for conditional probability is P(B|A) • Does not mean B is divided by A • It means that probability that event B occurs given that event A has already occurred

  10. Simple Example • Find the probability that an ace is picked from a deck of cards and is not replaced when the second card is picked and it is king. P(ace) * P(king without replacement)

  11. Multiplication Rule 2 • When two events are dependent of each other, the probability of both occurring is • P(A and B) = P(A) * P(B|A)

  12. Class Work • An urn contains 3 red balls, 2 blue balls and 5 white balls. A ball is selected and its color noted. Then it is replaced. A second ball is selected and its color noted. Find the probability of each of these. • Selecting 2 blue balls • Selecting 1 blue and then 1 white ball • Selecting 1 red ball and then 1 blue ball • A Harris poll found 46% of Americans say they suffer great stress at least once a week. If three people are selected at random, find the probability that all three will say that they suffer great stress at least once a weak. • A person owns a collection of 30 CDs, of which 5 are country music. If 2 CDs are selected at random, find the probability that b both are country music. • Three cards are drawn from an ordinary deck and not repalced. Find the probability of these: • Getting 3 jacks • Getting an ace, a king, and a queen in order • Getting a club, a spade, and a heart in order • Getting 3 clubs • Box 1 contains 2 red balls and 1 blue ball. Box 2 contains 3 blue balls and 1 red ball. A coin is toosed. If it falls heads up, box 1 is selected and a ball is drawn. If it falls tails up box 2 is selected and a ball is drawn. • Draw a tree diagram showing the situation • What is the sample space • Find the probability of selecting a red ball.

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