Probability of Independent Events

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Transcript Probability of Independent Events 

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 π‘Žπ‘›π‘‘ 𝐡 = 𝑃(𝐴) βˆ— 𝑃(𝐡)
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 π‘Žπ‘›π‘‘ 𝐡 π‘Žπ‘›π‘‘ 𝐢 = 𝑃 𝐴 βˆ— 𝑃 𝐡 βˆ— 𝑃(𝐢)
1 1 1
= βˆ— βˆ—
8 8 8
1
=
15
β‰ˆ .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 π‘›π‘œπ‘‘ β„Žπ‘’π‘Žπ‘Ÿπ‘–π‘›π‘” π‘ π‘œπ‘›π‘” =
=1βˆ’
Hearing your song on Monday and Tuesday are independent
𝑃 π»π‘’π‘Žπ‘Ÿπ‘–π‘›π‘” π‘†π‘œπ‘›π‘” = 1 βˆ’ [𝑃 π‘›π‘œπ‘‘ β„Žπ‘’π‘Žπ‘Ÿπ‘–π‘›π‘” π‘‘β„Žπ‘’ π‘ π‘œπ‘›π‘” ]
5
β‰ˆ 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 𝐡 𝐴
Probability of Dependent Events
If A and B are dependent events, then the probability
that both A and B occur is:
𝑃 𝐴 π‘Žπ‘›π‘‘ 𝐡 = 𝑃(𝐴) βˆ— 𝑃(𝐡|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
# π‘œπ‘“ π»π‘’π‘Ÿπ‘Ÿπ‘–π‘π‘Žπ‘›π‘’π‘ 
𝑃 π»π‘’π‘Ÿπ‘Ÿπ‘–π‘π‘Žπ‘›π‘’ =
π‘‡π‘œπ‘‘π‘Žπ‘™ πΆπ‘¦π‘π‘™π‘œπ‘›π‘’π‘ 
b.
=
760
1575
β‰ˆ .483
The probability that a future tropical cyclone in the Northern
Hemisphere is a hurricane.
# π‘œπ‘“ π»π‘’π‘Ÿπ‘Ÿπ‘–π‘π‘Žπ‘›π‘’π‘  𝑖𝑛 𝑁. π»π‘’π‘šπ‘–π‘ π‘β„Žπ‘’π‘Ÿπ‘’
=
π‘‡π‘œπ‘‘π‘Žπ‘™ πΆπ‘¦π‘π‘™π‘œπ‘›π‘’π‘  𝑖𝑛 𝑁. π»π‘’π‘šπ‘–π‘ π‘β„Žπ‘’π‘Ÿπ‘’
545
=
1142
β‰ˆ .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 π‘Žπ‘›π‘‘ 𝐡 = 𝑃 𝐴 βˆ— 𝑃(𝐡)
b.
39 13
=
βˆ—
52 52
3
=
16
β‰ˆ .188
You do not replace the first card?
𝑃 𝐴 π‘Žπ‘›π‘‘ 𝐡 = 𝑃 𝐴 βˆ— 𝑃 𝐡 𝐴
39 13
=
βˆ—
52 51
13
=
68
β‰ˆ .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, 𝐡 π‘Žπ‘›π‘‘ 𝐢 = 𝑃 𝐴 βˆ— 𝑃 𝐡 𝐴 βˆ— 𝑃 𝐢 𝐴 π‘Žπ‘›π‘‘ 𝐡
15 14 13
=
βˆ—
βˆ—
15 15 15
182
=
225
β‰ˆ .809
C=Friend Choses a Third Costume