Independent Events Independent events are two events in which the occurrence of one has no effect on the probability of the other.

Dependent Events Dependent events are two events in which the occurrence of one changes the probability of the other.

Carl Dick Heather Ellen Alan Greg

Probability of Dependent Events If A and B are dependent events, then P(A and B) = P(A) x P(B|A).

Probability of Dependent Events P(B|A) is read as “probability of B given A.”

Example 1 Select a name from the box and then select a second name without replacing the first. Find the probability of drawing a boy’s name followed by a girl’s name.

B = Select a boy’s name. G|B = Select a girl’s name, given that a boy’s name was selected on the first draw. 4 6 = 2 3 = 2 5 = P(B) P(G|B) P(B and G) = P(B) x P(G|B) 2 3 = x 2 5 4 15 = ≈ 0.27

Example 2 A bag of chocolate candies contains ten brown, eight orange, three yellow, and four green candies. What is the probability that the first two candies drawn from the bag without replacement will be brown?

B = Select a brown candy. B|B = Select a brown candy, given that a brown was already selected. 10 25 = 2 5 = 9 24 = 3 8 = P(B) P(B|B) P(B and B) = P(B) x P(B|B) 2 5 = x 3 8 3 20 = = 0.15

Example The names of ten club members, four boys and six girls, are placed in a hat. Jack is one of the boys and Sally is one of the girls. Suppose that the first name drawn will be the president and the second will be the vice-president.

Example Are the events independent or dependent? dependent

Example Find P(Jack, then a boy). 1 30

Example Find P(a girl other than Sally, then a boy). 2 9

Example Find P(a boy, then a girl). 4 15

Example Find P(a girl, then a boy). 4 15

Example What is the probability that one boy and one girl will be selected? 8 15

1 2 4 3

Probability of Independent Events If A and B are independent events, then P(A and B) = P(A) x P(B).

Example 3 Find P(4 and tails). 1 4 5 2

Find P(4 and tails). 2 6 = 1 3 = 1 2 = P(4) P(T) P(4 and T) = P(4) x P(T) 1 3 = 1 2 1 6 = ≈ 0.17

Example 4 A three-digit number is to be formed by drawing one of four slips of paper with the digits 1, 2, 3, and 4 from a hat. The first draw determines the first digit of the number to be formed, and so on.

Example 4 Digits can be used more than once, so the digit drawn is replaced in the hat before the next draw. What is the probability that the three-digit number formed is 123?

Find P(1 and 2 and 3). P(1 and 2 and 3) = P(1) x P(2) x P(3) 1 4 = x x 1 64 = ≈ 0.016

Example The names of ten club members, four boys and six girls, are placed in a hat. Jack is one of the boys and Sally is one of the girls. Suppose names will be drawn to select a boy’s representative and a girl’s representative.

Example Are the events independent or dependent? independent

Example What is the probability that Jack and Sally will be chosen as the representatives? 1 24

Example What is the probability that neither Jack nor Sally will be chosen? 5 8

Example What is the probability that Sally will be chosen but Jack will not? 1 8

Exercise In a Christian high school of 250 students, 92 play only the piano, 12 play only the trumpet, and 8 play both.

Exercise Use a Venn diagram to help you find the probability that each of the following will occur. Express your answer as both a fraction and a decimal rounded to the nearest thousandth.

Exercise Find the probability that a student drawn at random plays the trumpet. 2 25 = 0.08

Exercise Find the probability that a student drawn at random plays the piano. 2 5 = 0.4

Exercise Find the probability that a student drawn at random plays the piano and the trumpet. 4 125 = 0.032

Exercise Find the probability that a student drawn at random plays the piano, given that he plays the trumpet. 2 5 = 0.4

Exercise Does P(plays the piano and the trumpet) = P(plays the piano, given that he plays the trumpet)? no