The Multiplication Rule: Complements and Conditional Probability Section 4-5

What’s New? Probability of “at least 1” Formula for Conditional Probability

Challenge Problem You are considering purchasing 3 new TV’s for your college apartment. The size options at the store are 20”, 32” and 56” (assume there is an endless supply of all 3). If you randomly choose what sizes you will purchase, what is the probability that you don’t buy any 32” TV? What is the probability that you buy at least 1 32”? 1 – P(buy none)

Formula for “At Least 1” 𝑃 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 1 =1 −𝑃 𝑛𝑜𝑛𝑒 Getting at least 1 and getting none are complimentary events. Therefore their probabilities should sum up to 1. This is how we derive the above formula.

Example 1 Find the probability of a couple having at least 1 girl among 3 children. Assume that boys and girls are equally likely and that the gender of a child is independent of any other child. First, what is the probability of having no girls? Second, apply the complement rule! ½*1/2*1/2 = 1/8 1-1/8 = 7/8

Example 2 Assume that the probability of a defective Firestone tire is 0.0003. If the retail outlet CarStuff buys 100 Firestone tires, find the probability that they get at least 1 defective tire. (.9997)^100 = .970 1-.970 = .03

Use this time to relax or try the following rebus puzzles. Switching Gears … Use this time to relax or try the following rebus puzzles. All over again

Challenge Problem Use the table below to determine the probability that a person tests positive given that they actually lied. No (didn’t lie) Yes (lied) Positive Test 15 42 Negative Test 32 9 42/98 P(positive | lied) = P(positive and lied) / P(lied) = (42/98) * (98/51) = 42/51 = .824

What Do We Need To Know? Probability of an event is often affected by knowledge of circumstances. For example, the probability of a golfer making a hole in one is 0.000083 (based on past results). The probability of a professional golfer making a hole in one? 0.000421

Conditional Probability Using the Multiplication Rule, we can derive a new formula for calculating conditional probability. 𝑃 𝐵 𝐴 = 𝑃(𝐴 𝑎𝑛𝑑 𝐵) 𝑃(𝐴) Conditional probability of an event is a probability obtained with the additional information that some other event has already occurred.

Example 3 A math teacher gave her class two tests. 25% of the class passed both tests and 42% of the class passed the first test. What is the probability of passing the second test if the first test was passed? P(pass 2nd test | passed 1st test) = P (passed 1st and 2nd) / P(passed 1st) = .25/.42 = .595

Example 4 At Kennedy Middle School, the probability that a student takes Technology and Spanish is 0.087. The probability that a student takes Technology is 0.68. What is the probability that a student takes Spanish given that the student is taking Technology? What is the probability of getting at least one 5-answer multiple choice question correct out of 4 questions? P(Spanish | Technology) = P(Spanish and Technology) / P(Technology) = .087/.68 = .128 1 – P(none correct) = 1 – (4/5)*(4/5)*(4/5)*(4/5) = 1 - .410 = .590

Example 5 In New York State, 48% of all teenagers own a skateboard and 39% of all teenagers own a skateboard and roller blades. What is the probability that a teenager owns roller blades given that the teenager owns a skateboard? A company tests a batch of altimeters without replacement to see if they are acceptable for distribution. If one altimeter is faulty, then the whole batch is denied. Given that 3% of altimeters are faulty, what is the probability that a batch of 400 altimeters will be denied based on the first two items selected? P(owns roller blades | owns skateboard) = P(owns blades and skateboard)/ P(owns skateboard) = .39/.48 = .813 1 – P(1st and second aren’t faulty) = 1 - .97*.97 = 1 - .941 = .0590 ****this is the alarm clock problem

Example 6 The probability that it is Friday and that a student is absent is 0.03. Since there are 5 school days in a week, the probability that it is Friday is 0.2. What is the probability that a student is absent given that today is Friday? P(absent | Friday) = P(absent and Friday)/ P(Friday) = .03/.2 = .15

Homework p.175: 12, 15, 17, 19-22 Project Write-up due tomorrow