[Independent Probability, Conditional Probability, Tree Diagrams]


 Kenneth Parsons
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1 Name: Year 1 Review 119 Topic: Probability Day 2 Use your formula booklet! Page 5 Lesson 118: Probability Day 1 [Independent Probability, Conditional Probability, Tree Diagrams] Read and Highlight Station 1: Sample Space and Independent Events Independent Events: when one event doesn t affect the other. (i.e., Getting a 6 on a fair die and it raining outside.) When you KNOW they are independednt P(A B) = P(A)P(B) Sample space is a list of all possible outcomes Universal Set set of all possible outcomes (Notation is U) *Always use curly brackets { } when listing elements of a set Example) A die Guided Example 1: 1. In an experiment a coin is tossed and a die is rolled. a) Draw the sample space diagram for this experiment. b) Hence find the probability that in a single experiment you obtain a head and a number less than 3 on the die. c) Hence find the probability that in a single experiment you obtain a head or a number less than 3 on the die.
2 Guided Example 2: 2. Let P(A) = 0.6, P(B) = 0.5 and ( ). Are A and B independent? Justify your answer with mathematics. You Try! 3. An urn contains 3, 4 yellow and 5 black marbles. Three marbles are drawn at random without replacement. Find the probability a is drawn first, a yellow next, and a black is drawn last. 4. Consider events E and F where P(E) =.4, P(F) =.5, and ( ) =.2 Are E and F independent events? Show math to justify your answer.
3 Station 2: Conditional Probability Read and Highlight Conditional Probability **NEW** Notation To find the probability of the event A given the event B, we restrict our attention to the outcomes in B. We then find the fraction of those outcomes A that also occur. From the formula sheet As a picture Starting points for problems, when determining conditional probabilities:  If given the probabilities, just use the formula  If given a table, circle the column/row of what you are given and use that as your denominator Guided Example 3 5. Given that ( ) ( ) ( )  If you have a Venn Diagram, circle the set you are given and that is your denominator  If given data fill into either a table or a Venn Diagram You try! 6. For events A and B it is known that P(A B) ( ) ( ) a) P(A B) b) P(A B)
4 Read and Highlight Station 3: Tree diagrams When working with tree diagrams always remember to: 1. Always fill in all the values on your branches 2. Multiply across the branches! 3. ADD down the branches! 4. Use these products to get your answer! NEVER use the values on the branches! Guided Example 4 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 (a) Complete the tree diagram given below, showing all the probabilities. 0.1 FAULTY Probability outcomes 0.6 type A NOT FAULTY 0.4 type B FAULTY NOT FAULTY (b) A calculator is selected at random from the box. Find the probability that the calculator is (i) not faulty. (ii) Type B, given it was not faulty.
5 You Try! 7. At the basketball game, Amanda got into a twoshot foul situation. She figu her chance of making the first shot was 0.7. If she made the first shot, her chance of making the second shot was 0.6. However, if she missed the first shot, her probability of making the second shot was only 0.4. a) Complete the Tree Diagram above. b) What is the probability Amanda misses the second shot. c) Given Amanda missed the second shot, find the probability that she made the first shot. Station 4: Mixed Practice Complete each of the following problems. Show work and thinking for each. Use the formula booklet to help you through the problems. Don t Forget to Check the key tonight! 8. I toss a coin and roll a sixsided dice. Find the probability that I get a head on the coin and don t get a 6 on the dice? 9. Given that P(B) = 0.8 and P(A B) = 0.6. Find P(A B).
6 10. My wardrobe contains 7 shirts with one blue, one brown, two, one white and two black. I reach into the wardrobe and choose a shirt without looking to wear for work. Then I grab another shirt from the wardrobe for my cousins birthday dinner after work What is the probability that I will choose a shirt both times? 11. In the real estate ads, 64% of homes have garages, 21% have swimming pools, and 17% have both features. a) What s the probability a home has a garage or pool? b) Are taking these two events independent events? Justify with mathematics. 12. In a survey of 200 people, 90 of whom were female, it was found that 60 people were unemployed including 20 males. a) Using this information, complete the table below. Unemployed Employed Males Females Totals Totals 200 b) If a person is selected at random from this group of 200, find the probability that this person is i. an unemployed female; ii. a male, given that the person is employed.
7 13. Jim drives to work each day through two sets of traffic lights. The probability of the first set of traffic lights being is If the first set is then the probability that the next set of traffic lights is is If the first set is not, the probability that the next set is is not not a) Complete the tree diagram above. b) Calculate the probability that the second set of traffic lights is not c) Calculate the probability that the first light is not, given the second set of traffic lights is 14. Let P(A) = 0.6, P(B) = 0.5 and ( ) a) Draw a Venn diagram to represent the given information Find each of the following probabilities b) ( ) c) ( ) d) ( ) e) ( )
8 15. There are 27 Students in a class. 15 take art, 20 take theater and four take neither subject. How many students take both subjects? One person is chosen at random. Find the probability that he or she: a) takes theater but not art b) Takes at least one of the two subjects c) Take theater given that he or she takes art 16. Box A contains three cards bearing the numbers 1,2, 3. A second box, Box B, contains four cards with numbers 2,3,4,5. A card is chosen at random from each box. a) Draw the sample space diagram for the random experiment. Find the probability that: b) The cards have the same number c) The sum of the two numbers on the cards is a less than 7 d) A 2 is is drawn from A and a 3 is drawn from B. e) A 2 is is drawn from A or a 3 is drawn from B.
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