When do I Multiply Probabilities?
Example 1
Two Independent Events
In a game, a player must spin a spinner twice to determine the number of cards that they will draw.
The spinner has five equal–sized sections labeled 1 through 5. What is the probability that the
player spins an even number on the first spin and a 5 on the second spin?
Explore
Let A be the event that the first spin is an even number and let B be the event that the
second spin is a 5. The number on the first spin does not affect the number on the second
spin, so these events are independent.
Plan
Since there are 2 possibilities for even numbers, the probability that the first spin is even
is
is
Solve
2
5
1
5
. Since there is only one 5 on the spinner, the probability that the second spin is a 5
.
P(A and B) = P(A) P(B)
=
2
1
5
5
or
Probability of independent events
2
Substitute and multiply.
25
The probability that the first spin is an even number and the second spin is 5 is
Check
2
25
You can verify this result by making a tree diagram that includes the probabilities.
.
Example 2
Three Independent Events
In a game, you roll an octahedral die three times in a row to determine what will occur during your
turn. If you roll a 6 three times in a row, you lose your turn. What is the probability that you will
roll a six three times in a row? (An octahedral die has eight sides labeled 1 – 8.)
The probability of rolling a six on any roll is
1
8
P(three sixes) = P(six) P(six) P(six)
=
1
1
1
8
8
8
or
1
512
.
Probability of independent events
Substitute and multiply.
The probability of rolling a 6 three times in a row is
1
512
.
Example 3
Two Dependent Events
Sarah participated in a math contest at her school. She won 2 of the competitions. For each win, she
gets to select a card with the name of a prize that she will win. There are 15 cards in the bag that
she can choose from. Of the 10 cards, 1 has a TV, 3 have DVD players, 5 have CD’s, and 6 have gift
certificates. If she draws the cards at random and does not replace them, find the probability that
she draws a DVD player, then a CD.
Because the first card is not replaced, the events are dependent. Let D represent the DVD player, and C,
the CD.
P(D, then C) = P(D) P(C following D)
1
=
or
15 14
42
1
5
The probability of a DVD player then a CD is
Dependent events
After the first card is drawn, there are 14 left.
1
or about 2.4%.
42
Example 4
Three Dependent Events
Elise has a basket of thread she uses for craft projects. The basket contains 6 spools of white, 4
spools of black, 3 spools of red, 2 spools of gray, 2 spools of yellow, and 1 spool of green thread. If
she reaches into the basket and selects 3 spools of thread at random with no replacement, what is
the probability that she will first select a red spool, then a white spool, and then another white
spool?
Since the spools are not replaced, the events are dependent. Let R represent a red spool and W a white
spool.
P(R, W, W) = P(R) P(W following R) P(W following R and W)
=
3
6
5
18
17
16
or
5
272
As the spools are selected and not replaced, the total
number of spools decreases from 18 to 17 to 16. The
number of white spools decreases from 6 to 5 after the
first one is selected.
The probability is
5
272
or about 1.8%.