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

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

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 ( 𝐴𝑎𝑛𝑑𝐵 )=𝑃 (𝐴)∗𝑃 (𝐵)

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 ( 𝐴𝑎𝑛𝑑𝐵𝑎𝑛𝑑𝐶 )=𝑃 ( 𝐴)∗𝑃 (𝐵 )∗𝑃 (𝐶)

¿18∗18∗18

¿115 ≈ .00195

𝑃 (𝑛𝑜𝑡 h𝑒𝑎𝑟𝑖𝑛𝑔𝑠𝑜𝑛𝑔 )=

Hearing your song on Monday and Tuesday are independent

𝑃 (𝐻𝑒𝑎𝑟𝑖𝑛𝑔𝑆𝑜𝑛𝑔 )=1−[𝑃 (𝑛𝑜𝑡 h𝑒𝑎𝑟𝑖𝑛𝑔 h𝑡 𝑒𝑠𝑜𝑛𝑔 )]  5

¿1−( )5

≈0.763

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 ( 𝐴𝑎𝑛𝑑𝐵 )=𝑃 (𝐴)∗𝑃 (𝐵∨𝐴)

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 (𝐻𝑢𝑟𝑟𝑖𝑐𝑎𝑛𝑒 )=¿𝑜𝑓 𝐻𝑢𝑟𝑟𝑖𝑐𝑎𝑛𝑒𝑠𝑇𝑜𝑡𝑎𝑙𝐶𝑦𝑐𝑙𝑜𝑛𝑒𝑠

=≈ .483

¿¿𝑜𝑓 𝐻𝑢𝑟𝑟𝑖𝑐𝑎𝑛𝑒𝑠 𝑖𝑛𝑁 . h𝐻𝑒𝑚𝑖𝑠𝑝 𝑒𝑟𝑒𝑇𝑜𝑡𝑎𝑙𝐶𝑦𝑐𝑙𝑜𝑛𝑒𝑠 𝑖𝑛𝑁 . h𝐻𝑒𝑚𝑖𝑠𝑝 𝑒𝑟𝑒

¿5451142≈ .477

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 ( 𝐴𝑎𝑛𝑑𝐵 )=𝑃 ( 𝐴)∗𝑃 (𝐵)¿3952∗1352¿316≈ .188

𝑃 ( 𝐴𝑎𝑛𝑑𝐵 )=𝑃 ( 𝐴)∗𝑃 (𝐵|𝐴 )¿3952∗1351

¿1368≈ .191

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 ,𝐵𝑎𝑛𝑑𝐶 )=𝑃 ( 𝐴 )∗𝑃 (𝐵|𝐴 )∗𝑃 (𝐶|𝐴𝑎𝑛𝑑𝐵 )

¿1515∗1415

∗1315

¿182225≈ .809