Rita Korsunsky

Use a tree diagram to solve these problems

Example With fluWithout flu highlowhighlow During a flu epidemic, 35% of a school’s students have the flu. Of those with the flu, 90% have high temperatures. However, a high temperature is also possible for people without the flu; in fact, the school nurse estimates that 12% of those without the flu have high temperatures. Incorporate the facts given above into a tree diagram.

During a flu epidemic, 35% of a school’s students have the flu. Of those with the flu, 90% have high temperatures. However, a high temperature is also possible for people without the flu; in fact, the school nurse estimates that 12% of those without the flu have high temperatures. b. About what percent of the student body have a high temperature? With flu Without flu highlow highlow There are 2 paths (one OR the other) that lead to a T (high temperature)

Example 1 } { 1 Y Y Y B B B & are placed in a jar. One ball is chosen, the color is noted and it is placed back in the jar. This is repeated for the 2 nd ball. What is the probability that both balls are the same color? Probability that both balls are the same color =

Example 2 & are placed in a jar. One ball is chosen, the color is noted and the ball is not replaced. What will be the probability that the two balls are the same color? Probability that both balls are the same color = Y Y Y B B B

The probability that the second ball is blue does NOT depend on the color of the first ball, therefore the two events are called independent events. In general, two events C and D are independent if and only if Conditional Probability equal We denote the probability that the second ball is yellow given that the first ball is yellow by: P(second is Y|first is Y) This is called conditional probability. (with replacement) Y Y Y B B BY Y Y B B B (without replacement) The probability that the second ball is blue DOES depend on the color of the first ball, therefore the two events are called dependent events Not equal

Ex. 3: A card is randomly drawn from a standard deck. a. Show that the events “jack” and “spade” are independent. Since they are equal, the events “jack” and “spade” are independent. Probability of Events Occurring Together b. Show how Rule 2 can be used to find the probability of drawing the jack of spades.

A B

Each student in a class of 30 studies one foreign language and one science. Chem(C)Physics(P)Bio(B)Totals French(F)74314 Spanish(S)16916 Totals a. Find the probability that a randomly chosen student studies Chemistry. b. Find the probability that a randomly chosen student studies Chemistry given that the student studies French. c. Are the events “students study Chemistry” and “students study French” independent? Example 4

EMPTY SLIDE