Advanced Algebra Notes
Section 10.5: Find Probabilities of
Independent and Dependent
Events
Two events are independent
if the
occurrence of one has no effect on the
occurrence of the other. For instance, if a coin
is tossed twice, the outcome of the first toss
(heads or tails) has no effect on the outcome of
the second toss.
Probability of Independent Events
If A and B are independent events, then the probability
that both A and B occur is: 𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃(𝐴) ∗ 𝑃(𝐵)
More generally, the probability that n independent
events occur is the product of the n probabilities of the
individual events.
Example 1
In a BMX meet, each heat consists of 8 competitors who are randomly assigned
lanes from 1 to 8. What is the probability that a racer will draw lane 8 in the 3
heats in which the racer participates?
A is first heat
B is second heat
C is third heat
𝑃 𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑛𝑑 𝐶 = 𝑃 𝐴 ∗ 𝑃 𝐵 ∗ 𝑃(𝐶)
1 1 1
= ∗ ∗
8 8 8
1
=
15
≈ .00195
Example 2
While you are riding to school, your portable CD player randomly plays 4
different songs from a CD with 16 songs on it. What is the probability that you
will hear your favorite song on the CD at least once during the week (5 days)?
𝑃 𝑛𝑜𝑡 ℎ𝑒𝑎𝑟𝑖𝑛𝑔 𝑠𝑜𝑛𝑔 =
=1−
Hearing your song on Monday and Tuesday are independent
𝑃 𝐻𝑒𝑎𝑟𝑖𝑛𝑔 𝑆𝑜𝑛𝑔 = 1 − [𝑃 𝑛𝑜𝑡 ℎ𝑒𝑎𝑟𝑖𝑛𝑔 𝑡ℎ𝑒 𝑠𝑜𝑛𝑔 ]
5
≈ 0.763
5
Two events A and B are dependent events
if the
occurrence of one affects the occurrence of the other.
The probability that B will occur given that A has
conditional events
occurred is called the
of B
given A is written as 𝑃 𝐵 𝐴
Probability of Dependent Events
If A and B are dependent events, then the probability
that both A and B occur is:
𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃(𝐴) ∗ 𝑃(𝐵|𝐴)
Example 4
The table shows the numbers of tropical cyclones that formed during the
hurricane seasons from 1988 to 2004. Use the table to estimate
a.
The probability that a future tropical cyclone is a hurricane
# 𝑜𝑓 𝐻𝑢𝑟𝑟𝑖𝑐𝑎𝑛𝑒𝑠
𝑃 𝐻𝑢𝑟𝑟𝑖𝑐𝑎𝑛𝑒 =
𝑇𝑜𝑡𝑎𝑙 𝐶𝑦𝑐𝑙𝑜𝑛𝑒𝑠
b.
=
760
1575
≈ .483
The probability that a future tropical cyclone in the Northern
Hemisphere is a hurricane.
# 𝑜𝑓 𝐻𝑢𝑟𝑟𝑖𝑐𝑎𝑛𝑒𝑠 𝑖𝑛 𝑁. 𝐻𝑒𝑚𝑖𝑠𝑝ℎ𝑒𝑟𝑒
=
𝑇𝑜𝑡𝑎𝑙 𝐶𝑦𝑐𝑙𝑜𝑛𝑒𝑠 𝑖𝑛 𝑁. 𝐻𝑒𝑚𝑖𝑠𝑝ℎ𝑒𝑟𝑒
545
=
1142
≈ .477
Example 5
You randomly select two cards from a standard deck of 52
cards. That is the probability that the first card is not a heart
and the second is a heart if:
a.
You replace the first card before selecting the second
𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃 𝐴 ∗ 𝑃(𝐵)
b.
39 13
=
∗
52 52
3
=
16
≈ .188
You do not replace the first card?
𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃 𝐴 ∗ 𝑃 𝐵 𝐴
39 13
=
∗
52 51
13
=
68
≈ .191
Example 6
You and two friends go to the same store at different
times to buy costumes for a costume party. There are
15 different costumes at the store, and the store has at
least 3 duplicates of each costume. What is the
probability that you each choose different costumes?
A= Costume
B= Different Costume Chosen
𝑃 𝐴, 𝐵 𝑎𝑛𝑑 𝐶 = 𝑃 𝐴 ∗ 𝑃 𝐵 𝐴 ∗ 𝑃 𝐶 𝐴 𝑎𝑛𝑑 𝐵
15 14 13
=
∗
∗
15 15 15
182
=
225
≈ .809
C=Friend Choses a Third Costume