4. Are events C and D independent? Verify your answer with a calculation.

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1 Honors Math 2 More Conditional Probability Name: Date: 1. A standard deck of cards has 52 cards: 26 Red cards, 26 black cards 4 suits: Hearts (red), Diamonds (red), Clubs (black), Spades (black); 13 of each suit Numbered cards 2-10 in each suit (so, four of each number) A, Ace is usually valued at 1, and there are 4 in the deck (one of each suit) Face cards are J Jack (11), Q Queen (12), K King (13) and there are 4 of each in the deck, one in each suit (12 total) Trial: Select one card from the deck. Event A: card if a Jack Event C: card is red Event B: card is black Event D: card is a heart 1. Find the probability of each event: P(A) = P(B) = P(C) = P(D) = 2. Find the following probabilities (THINK don t use a formula!) P(A and B) = P(C and D) = 3. Are events A and B independent? Verify your answer with a calculation. 4. Are events C and D independent? Verify your answer with a calculation. 5. Come up with another pair of events for the trial draw one card from a deck that are independent. 6. Come up with another pair of events for the trial draw one card from a deck that are mutually exclusive.

2 Using Venn Diagrams students were asked if they take Math and History classes. The Venn diagram shows the number of students who took each class. Suppose one student is chosen at random, and answer the questions below. Region A: math Region B: history A B a. What does the 17 represent in the diagram above? b. P(student takes Math) = c. P(student takes History) = d. P(student takes Math AND History) = e. P(student takes Math OR History) = f. Are taking math and taking history independent? Verify your answer with a calculation.

3 Drawing Tree Diagrams to find probabilities To picture the probabilities in situations involving multiple events, it is helpful to make a tree diagram showing all the possible sequences of events with their probabilities. 3. In a deck of 52 playing cardes, 13 of the cards are hearts (H) and the other 39 cards are non-hearts (N). The diagram represents dealing one card from this deck and then dealing a second card from the deck. Deal first card H N Deal second card H N H N a. Use the tree diagram above to fill in the blanks of these statements!"!" is the probability of!"!" is the probability of given that!"!" is the probability of given that b. Explain the reasoning for the probability!"!" shown in the tree diagram. Why 38? Why 51? c. What is the probability that the second card is a heart given that the first card is a heart? d. What is the probability that the second card is a heart and the first card is a heart?

4 Making a Two-Way Table 4. Students records indicate that the probability of passing photography is 0.75, the probability of failing economics is 0.65, and the probability of passing at least one of the two courses is Complete the two- table below and the Venn diagram, then answer the questions. Passed Photography Failed Photography Total Passed Economics Failed Economics Total Region B: Passing only photography Region D: Passing only economics A B C D a. P(passing both courses) = b. P(failing both courses) = c. P(passing exactly one course) = d. P pass photography failed economics e. Are passing photography and passing economics independent?

5 women who recently gave birth were given a survey with 3 yes/no questions, and the following information was recorded. The mother Number of Respondents Is over 35 years of age 27 Just had her first child 21 Has a career 42 Is over 35 and just had first child 17 Has a career and just had first child 9 Is over 35 and has a career 16 Is over 35, has a career, and just had first child 7 Use the Venn Diagram to organize the information. If a mother from this group is randomly selected, find the probabilities below. a. P(replied yes to exactly 2 of the 3 questions above) b. P(replied yes to at least 2 of the 3 questions above) c. P(replied yes to at least one of the 3 questions above) d. P(replied yes to none of the 3 questions above)

6 6. Suppose that a jar contains 5 yellow marbles and 5 blue marbles. Two marbles are drawn from the jar (and not returned to the jar). a. Make a tree diagram. Label every branch with a probability. b. What is the probability that both marbles are yellow? c. What is the probability that both marbles are blue? d. What is the probability of drawing a blue marble followed by a yellow marble? e. What is the probability of drawing a yellow marble followed by a blue marble?

7 7. Use the info below and answer each question: There are 158 math majors at LJM University, enrolled as follows: 82 have enrolled in Linear Algebra, 61 have enrolled in Number Theory, and 40 have enrolled in Calculus. 14 are taking both Linear Algebra and Number Theory 18 are taking Linear Algebra and Calculus 11 are taking Number Theory and Calculus 8 brave souls taking all three maths a. Create a Venn Diagram to represent the information b. Find the probability that a student is not enrolled in any of the three courses c. Find the probability that a student is taking at least 2 courses d. Find the probability that a student is taking at most 2 courses e. Find the probability that a student is taking only one course f. Find the probability that a student is taking Calculus and Linear Algebra g. Find the probability that a student is taking Linear Algebra given that they are in Calculus h. Find the probability that a student is taking Calculus given that they are in Linear Algebra. i. Find the probability that a student is taking Calculus or Linear Algebra j. Are Calculus and Linear Algebra independent events? Show a calculation to justify your response. k. Give an example of two events from this set up that are mutually exclusive.

8 8. For a standard deck of cards, draw one card (this is the trial). Answer each question. a. Draw a two- way table to represent the events black and Jack. (think about black and not black with Jack and not jack ) b. Recall that these events (black and Jack) are independent. Explain how the table shows this. c. Draw a two- way table to represent the events red and heart. d. Recall that Red and Heart are dependent events. Explain how the table shows this.

9 9. Every year, a large company surveys a representative sample of its employees. Among the survey questions are, Do you exercise regularly? and Do you usually feel alert or tired at work? The company wants to conduct an analysis of the data to see if encouraging employees to exercise may increase their alertness on the job. Of the employees surveyed, 65% exercise regularly. Of those who exercise regularly, 81% feel alert during the workday. Of those who do not exercise regularly, 69% feel alert during the workday. 1. What percentage of the employees feel alert? 2. What percentage of employees who exercise regularly feel alert? 3. Suppose you select an employee from the survey at random. Is selecting an employee who feels alert during the workday independent of selecting an employee who exercises regularly? Explain. 4. What would the distributions have to look like to make those two events independent? Re-draw the table with frequencies that result in the events Alert and Exercise being independent.