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Name_________________________________ Date: ____________ Lesson 8-7: Conditional Probability Learning Goals: #8: What is conditional probability? How do we determine a conditional probability? Conditional Probability In words: The probability of an event (A), given that another event (B) has already occurred. In symbols: π( π΄| π΅) = ο· π(π΄ and π΅) π(π΅) (given in formula booklet) Restrict the total outcomes to be the outcomes of B. Then find the probability of both events occurring. Example: In a school of 1200 students, 250 are seniors, 150 students take math, and 40 students are seniors and are also taking math. What is the probability that a randomly chosen student is taking math, given they are a senior? Letβs try an example using information from our class: P(Boy|Boots)= Male Boots P(Girl|Boots)= No Boots Total P(Boots|Girl)= Female Total Finding Conditional Probability 2 2 1) Given that π(πΆ) = 3 πππ π(πΆ β© π·) = 5 , ππππ π(π·|πΆ). 2) At a school, 60% of students buy a school lunch. Only 10% of students buy lunch and dessert. What is the probability that a student who buys lunch also buys dessert? 3) Let P(A) = 0.6, P(B) = 0.5 and π(π΄ β© π΅) = 0.2 a. Draw a Venn diagram to represent the given information b. Find each of the following probabilities: i. π(π΄ βͺ π΅) ii. π((π΄ βͺ π΅)β² ) iii. π(π΄|π΅) iv.π(π΅|π΄) 4) Given that P(B) = .8 and P(A β© π΅) = 0.6, ππππ π(π΄|π΅). 5) 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? Finding Conditional Probability from Tables 6) Researchers analyzed 200 adults to determine if there was a link between their highest level of education and whether or not they smoked. Use the table at the right to answer the following questions, leave all questions in fraction form and use proper notation. a) Given the person smoked, what's the probability they graduated high school? b) Given the person smoked, what's the probability they went to a 2 year college? 7) In a survey, 100 students were asked βdo you prefer to watch television or play sports?β Of the 46 boys in the survey, 33 said they would choose sports, while 29 girls made this choice. Boys Girls Total Television 13 25 38 Sport 33 29 62 Total 46 54 100 By completing this table or otherwise, find the probability that a) a student selected at random prefers to watch television; b) a student prefers to watch television, given that the student is a boy. Finding Conditional Probability from Tree Diagrams 8) The events B and C are dependent, where C is the event βa student takes Chemistryβ, and B is the event βa student takes Biologyβ. It is known that the probability a student takes Chemistry is 0.4. For students that take Chemistry the probability they also take Biology is 0.6, otherwise the probability they take Biology is 0.5. a) Complete the following tree diagram. Chemistry Biology Remember: B 0.4 C Bο’ In a tree diagram you must multiply along the branches to determine probabilities! B Cο’ Bο’ b) Calculate the probability that a student takes Biology. c) Given that a student takes Biology, what is the probability that the student takes Chemistry? 9) A teacher has a box containing six type A calculators and four type B calculators. The probability that a type A calculator is faulty is 0.1 and the probability that a type B calculator is faulty is 0.12. a) Complete the tree diagram given below, showing all the probabilities. 0.1 FAULTY type A 0.6 NOT FAULTY FAULTY 0.4 type B NOT FAULTY b) Given you pick a not faulty calculator, whatβs the probability itβs type A? Practice: 10) The table below shows the number of left and right handed tennis players in a sample of 50 males and females. Left handed Right handed Total Male 3 29 32 Female 2 16 18 Total 5 45 50 If a tennis player was selected at random from the group, find the probability that the player is (a) male and left handed; (b) right handed; (c) right handed, given that the player selected is female. 11) A quality-control inspector checks for defective parts. The table shows the results of the inspectorβs work. Find: (a) the probability that a defective part βpassesβ (b) the probability that a non-defective part βfailsβ 12) A bag contains two red sweets and three green sweets. Jacques takes one sweet from the bag, notes its colour, then eats it. He then takes another sweet from the bag. a) Complete the tree diagram below to show all probabilities. Red 2 5 Red Green Red 2 4 Green Green b) Given that he chooses a red sweet from the bag first, what is the probability he will choose another red sweet? c) Find the probability of Jacques choosing a green sweet, given that he has already chosen a green sweet. 13) Lizzie is attempting two exam questions. 2 The probability that she gets any exam question correct it 3. a) Fill in the tree diagram. a) What is the probability that Lizzie will get the second one correct, given that she answers the first question incorrect? b) Given that Lizzie gets the first one correct, what is the probability that she will get the second one correct?