Lesson 12-5
Real-World Example 1 Independent Events
MARBLES A bag contains 12 marbles. Three are green, two are blue, 6 are yellow and 1 is
red. Jama chooses one marble, records the color, and places it back into the bag. She then
chooses another marble. Find the probability that Jama chooses a green marble each time.
Since the first marble is replaced, this choice does not affect the selection of the second marble.
3 number of green marbles
First marble: P(green) = 12
total number of marbles
3 number of green marbles
Second marble: P(green) = total number of marbles
12
P(A and B) = P(A) P(B) Probability of independent events
P(green and green) = P(green) P(green)
3 3
= Substitution
12 12
9
= 144 Multiply.
The probability that Jama will choose a green marble each time is 6.25%.
Example 2 Dependent Events
CARDS Four cards are drawn randomly from a standard deck of cards and not replaced.
Find each probability if the cards are drawn in the order indicated.
a. P(four, king, queen)
The selection of the first card affects the selection of the next card since there is one less card from
which to choose. So, the events are dependent.
4 1 number of fours
First card: P(four) = or
52 13 total number of cards
4 number of kings
Second card: P(king) =
51 number of cards remaining
4 2 number of queens
Third card: P(queen) = or
50 25 number of cards remaining
P(four, king, queen) = P(four) P(king) P(queen)
1 4 2
= Substitution.
13 51 25
8
= Multiply.
16,575
8
The probability is .
16,575
b. P(spade, club, not heart)
Since the card that is not a heart is selected after the first two cards, there are 39 – 2 or 37 cards that
are not hearts.
P(spade, club, not heart) = P(spade) P(club) P(not heart)
13 13 37
= Substitution
52 51 50
6253 481
= or Multiply.
132, 600 10, 200
481
The probability is or about 0.5%.
10, 200
Real-World Example 3 Mutually Exclusive Events
MUSIC Brad has 25 CDs. Eight are rock, eight are jazz, two are rap and seven are
classical. He chooses one CD at random. Find each probability.
a. P(jazz or rap)
Since the CD cannot be both a jazz and a rap CD, the events are mutually exclusive.
8 number of jazz CDs
P(jazz) =
25 total number of CDs
2 number of rap CDs
P(rap) =
25 total number of CDs
P(jazz or rap) = P(jazz) + P(rap) Definition of mutually exclusive events
8 2
= + Substitution
25 25
10 2
= or Add.
25 5
2
The probability of choosing a jazz CD or a rap CD is or 40%.
5
b. P(rock or classical)
8 number of rock CDs
P(rock) =
25 total number of CDs
7 number of classical CDs
P(classical) =
25 total number of CDs
P(rock or classical) = P(rock ) + P(classical) Definition of mutually exclusive events
8 7
= + Substitution
25 25
15 3
= or Add.
25 5
3
The probability of choosing a rock CD or a classical CD is or 60%.
5
Real-World Example 4 Events that are Not Mutually Exclusive
STUDENTS There are 26 students in Mr. Collins’ English class. Twelve are seniors and
fourteen are juniors. Eight of the seniors are boys and six of the juniors are boys. What is
the probability that a randomly selected student is a boy or a senior?
Since some students are boys and seniors, the events are not mutually exclusive.
14 12 8
P(boy) = P(senior) = P(boy and senior) =
26 26 26
P(boy or senior) = P(boy) + P(senior) – P(boy and a senior)
14 12 8
= + – Substitution.
26 26 26
18 9
= or Simplify.
26 13
9
The probability of a student being a boy or a senior is or about 69%.
13