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Advanced Algebra Notes Section 10.5: Find Probabilities of Independent and Dependent Events

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Page 1: Advanced Algebra Notes Section 10.5: Find Probabilities of Independent and Dependent Events

Advanced Algebra NotesSection 10.5: Find Probabilities of Independent and Dependent Events

Page 2: Advanced Algebra Notes Section 10.5: Find Probabilities of Independent and Dependent Events

Two events are 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.

independent

Page 3: Advanced Algebra Notes Section 10.5: Find Probabilities of Independent and Dependent Events

Probability of Independent Events If A and B are independent events, then the probability that both A and B occur is: More generally, the probability that n independent events occur is the product of the n probabilities of the individual events.

𝑃 ( π΄π‘Žπ‘›π‘‘π΅ )=𝑃 (𝐴)βˆ—π‘ƒ (𝐡)

Page 4: Advanced Algebra Notes Section 10.5: Find Probabilities of Independent and Dependent Events

Example 1In 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?

Example 2While 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)?

B is second heat C is third heat

𝑃 ( π΄π‘Žπ‘›π‘‘π΅π‘Žπ‘›π‘‘πΆ )=𝑃 ( 𝐴)βˆ—π‘ƒ (𝐡 )βˆ—π‘ƒ (𝐢)

ΒΏ18βˆ—18βˆ—18

ΒΏ115 β‰ˆ .00195

𝑃 (π‘›π‘œπ‘‘ hπ‘’π‘Žπ‘Ÿπ‘–π‘›π‘”π‘ π‘œπ‘›π‘” )=

Hearing your song on Monday and Tuesday are independent

𝑃 (π»π‘’π‘Žπ‘Ÿπ‘–π‘›π‘”π‘†π‘œπ‘›π‘” )=1βˆ’[𝑃 (π‘›π‘œπ‘‘ hπ‘’π‘Žπ‘Ÿπ‘–π‘›π‘” h𝑑 π‘’π‘ π‘œπ‘›π‘” )]  5

ΒΏ1βˆ’( )5

β‰ˆ0.763

Page 5: Advanced Algebra Notes Section 10.5: Find Probabilities of Independent and Dependent Events

Two events A and B are if the occurrence of one affects the occurrence of the other. The probability that B will occur given that A has occurred is called the of B given A is written as Probability of Dependent Events If A and B are dependent events, then the probability that both A and B occur is:

dependent events

conditional events

𝑃 ( π΄π‘Žπ‘›π‘‘π΅ )=𝑃 (𝐴)βˆ—π‘ƒ (𝐡∨𝐴)

Page 6: Advanced Algebra Notes Section 10.5: Find Probabilities of Independent and Dependent Events

Example 4The 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

b. The probability that a future tropical cyclone in the Northern Hemisphere is a hurricane.

𝑃 (π»π‘’π‘Ÿπ‘Ÿπ‘–π‘π‘Žπ‘›π‘’ )=ΒΏπ‘œπ‘“ π»π‘’π‘Ÿπ‘Ÿπ‘–π‘π‘Žπ‘›π‘’π‘ π‘‡π‘œπ‘‘π‘Žπ‘™πΆπ‘¦π‘π‘™π‘œπ‘›π‘’π‘ 

=β‰ˆ .483

ΒΏΒΏπ‘œπ‘“ π»π‘’π‘Ÿπ‘Ÿπ‘–π‘π‘Žπ‘›π‘’π‘  𝑖𝑛𝑁 . hπ»π‘’π‘šπ‘–π‘ π‘ π‘’π‘Ÿπ‘’π‘‡π‘œπ‘‘π‘Žπ‘™πΆπ‘¦π‘π‘™π‘œπ‘›π‘’π‘  𝑖𝑛𝑁 . hπ»π‘’π‘šπ‘–π‘ π‘ π‘’π‘Ÿπ‘’

ΒΏ5451142β‰ˆ .477

Page 7: Advanced Algebra Notes Section 10.5: Find Probabilities of Independent and Dependent Events

Example 5You 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 b. You do not replace the first card?

𝑃 ( π΄π‘Žπ‘›π‘‘π΅ )=𝑃 ( 𝐴)βˆ—π‘ƒ (𝐡)ΒΏ3952βˆ—1352ΒΏ316β‰ˆ .188

𝑃 ( π΄π‘Žπ‘›π‘‘π΅ )=𝑃 ( 𝐴)βˆ—π‘ƒ (𝐡|𝐴 )ΒΏ3952βˆ—1351

ΒΏ1368β‰ˆ .191

Page 8: Advanced Algebra Notes Section 10.5: Find Probabilities of Independent and Dependent Events

Example 6You 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 C=Friend Choses a Third Costume

𝑃 ( 𝐴 ,π΅π‘Žπ‘›π‘‘πΆ )=𝑃 ( 𝐴 )βˆ—π‘ƒ (𝐡|𝐴 )βˆ—π‘ƒ (𝐢|π΄π‘Žπ‘›π‘‘π΅ )

ΒΏ1515βˆ—1415

βˆ—1315

ΒΏ182225β‰ˆ .809


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