Compound Probability

Yesterday, we worked with selecting one item… let’s review. Mr. Niceguy brought in a dozen donuts for the math department – there are 6 glazed, 4 chocolate, and 2 jelly donuts in the box. You reach in and grab one. Find the probability of picking a) A chocolate b) Not a jelly a) A chocolate or a glazed

1. What is the probability that a person likes Wendy’s? Or the data can be in a two-way table. 100 people were surveyed for their favorite fast-food restaurant. 1. What is the probability that a person likes Wendy’s? 2. What is the probability that a person is male who likes Burger King? 3. What is the probability that a person is likes McDonald’s or Burger King? McDonald’s Burger King Wendy’s Male 20 15 10 Female 25

Today we’re working with compound events… we’ll be selecting more than one. Examples: rolling a die and tossing a penny spinning a spinner and drawing a card tossing two dice tossing two coins

A compound event combines two or more events, using the word and or the word or. Two events are independent if the occurrence of one event has no effect on the other Two events are dependent if the occurrence of one event affects the outcome of the other

Independent Events P(A and B) = P(A) ∙ P(B) Two events A and B, are independent if A occurs & does not affect the probability of B occurring. Examples- Landing on heads from two different coins, rolling a 4 on a die, then rolling a 3 on a second roll of the die. Probability of A and B occurring: P(A and B) = P(A) ∙ P(B)

Example 1 A jar contains three red, five green, two blue and six red marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. What is the probability of choosing a green and a red marble? P (green) = 5/16 P (red) = 6/16 P (green and red) = P (green) ∙ P (red) = 15 / 128

Dependent Events Two events A and B, are dependent if A occurs & affects the probability of B occurring. Examples- Picking a blue marble and then picking another blue marble if I don’t replace the first one.

Example 2 A random sample of parts coming off a machine is done by an inspector. He found that 5 out of 100 parts are bad on average. If he were to do a new sample, what is the probability that he picks a bad part and then picks another bad part if he doesn’t replace the first? P (1st bad) = 5/100 P (2nd bad) = 4/99 P (bad and then bad) = 5/100 * 4/99 = 1/495

Example 3 A jar contains three red, five green, two blue and six yellow marbles. A marble is chosen at random from the jar. A second marble is chosen. What is the probability of choosing a green and a yellow marble if the first marble is not replaced? P (green) = 5/16 P (yellow) = 6/15 P (green and yellow) = P (green) ∙ P (yellow) = 30 / 240 = 1/8

Example 4 A jar contains three red, five green, two blue and six yellow marbles. A marble is chosen at random from the jar. A second marble is chosen. What is the probability of choosing a green marble both times if the first marble is not replaced? P (green) = 5/16 P (2nd green) = 4/15 P (green and green) = P (green) ∙ P (green) = 20 / 240 = 1/12

mutually exclusive overlapping independent dependent P(A or B) = P(A) + P(B) P(A or B) = P(A) + P(B) - P(overlap) -Drawing a king or a diamond -rolling an even sum or a sum greater than 10 on two dice -Selecting a female from Georgia or a female from Atlanta -Drawing a king or a queen -Selecting a male or a female -Selecting a blue or a red marble independent dependent P(A and B) = P(A) ∙ P(B) P(A and B) = P(A) ∙ P(B given A) WITHOUT REPLACEMENT: -Drawing a king and a queen -Selecting a male and a female -Selecting a blue and a red marble WITH REPLACEMNT: -Drawing a king and a queen -Selecting a male and a female -Selecting a blue and a red marble