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# Math 1620 Week 2 MATH 1620

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This 10 page Class Notes was uploaded by Mariah Figueiredo on Sunday February 7, 2016. The Class Notes belongs to MATH 1620 at Rensselaer Polytechnic Institute taught by Wing Sze E Kam in Spring 2016. Since its upload, it has received 96 views. For similar materials see Contemporary Mathematical Ideas in Society in Mathematics (M) at Rensselaer Polytechnic Institute.

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Date Created: 02/07/16

Week2 11.5-11.6 Probability Spring2016 Recall : Permutation Combination Order MATTERS!! DOES NOT MATTER Formula n ! nPr n ! nPr▯ nC r ▯ ▯n▯r ! r ! r ! n▯r ! Probabilityof anevent, P E ▯ number of outcomes in E ; total number of possible outcomes Example 1 (Probability withpermutation) There are 6 stand-up comics, A, B, C, D, E and F, are to perform ona single eveningat a comedyclub. The order ofperformance is determined byrandom selection. Total number ofpermutations = Find the probability that a. Comic E will perform first. b. Comic C will perform fifthand comic B will perform last. 1 c. The comedians will perform in the order D, E, C, A, B, F. d. Comic A or comic B will perform first. Example 2 (Probability withcombinations) A poker hand consists of5 cards is dealt from a shuffled deck of52 cards. a. Find the total number of possible 5-card poker hands. b. A diamond flush is a 5-card hand consisting of all diamond s. Find the number of possible diamond flushes. c. Find the probabilityof beingdealt a diamond flush. 2 d. Find the probabilityof beingdealt a flush (canbe heart, club, diamond or spade). Example 3 3 cards are dealt from a shuffled deck of52 cards, find the probability that all3 cards are picture cards. Example 4 4 cards are dealt from a shuffled deck of 52 cards, find the probability of getting2 queens and 2 kings. Q: What are mutually exclusive events? A: Ifevents A and B cannot occur simultaneously, the events are mutually exclusive. IfA and B are mutually exclusive, then P A or B ▯ P A ▯ P B▯ ▯ 3 Example 5 A card is dealt from a deck of 52 cards randomly. What is the probability of selectinga king or a queen? Q: What if the events are not mutually exclusive? A: IfA and B are NOT mutually exclusive, then P A or B ▯ P A ▯ P B ▯ P A and B ▯ Example 6 A card is dealt from a deck of 52 cards randomly. What is the probability of getting a 7 or a red card? 4 Suggested HW 11.5 :# 15, 18, 20 11.6 :# 28, 29, 30, extra 11.5 #15. A poker hand consists of5 cards. a. Find the total number of possible 5-card poker hands. b. A club flush is a 5-card hand consisting of all clubs. Find the number of possible club flushes. c. Find the probability of beingdealt a club flush. #18. 4 cards are dealt from a shuffled deck of 52 cards, find the probability that all 4 are hearts. #20. 4 cards are dealt froma shuffled deck of52 cards, find the probabilityof getting3 jacks and 1 queen. 11.6 #28. A card is dealt from a deck of 52 cards randomly. What is the probability of getting a 5 or a black card? #29. A card is dealt from a deck of 52 cards randomly. What is the probability of getting a heart or a picture card? #30. A card is dealt from a deck of 52 cards randomly. What is the probability of getting a card > 2 and < 7, or a diamond? #extra. In a 5-card poker hand, find the probability of being dealt Full House (3 cards of one rank and 2 cards of a second rank) 5 Week2 11.7-11.8 Multiplication Rule, Conditional Probability, ExpecteSpring 2016 Independent events: Event A and event B are independent if the occurrence of either of A or B has no effect on the probability of the other, e.g. toss a coin or roll a die many times in succession. If A and B are independent events, then P A and B ▯ P A ▯P B ▯ (multiplication rule) Example 1 5 If the probability that South Florida will be hit by a hurricane in any single yea, is 19 find the probability that South Florida a. will be hit by a hurricane in 4 consecutive years? Saffir/Simpson hurricane scale Category Winds (miles/hour) 1 74-95 2 96-110 3 111-130 4 131-155 5 >155 b. will NOT be hit by a hurricane in the next 4 years? c. will be hit by a hurricane at least once in the next 4 years? 1 Q: What if A and B are dependent? A: If A and B are dependent events, then P A and B ▯ P A ▯P B given that A has occured ▯ It can be written as P A and B ▯ P A ▯P B | A ▯ Example 2 TWO cards are being dealt from a 52-card deck. Find the probability of getting 2 kings. Example 3 Shared birthdays a. If 3 people are selected at random, find the probability that they all have different birthdays. b. If 3 people are selected at random, find the probability that at least 2 of them have the same birthday. 2 Example 3 (cont’) c. If 23 people are selected at random, find the probability that at least 2 of them have the same birthday. Conditional probability number of outcomes common to B and A P B | A ▯ number of outcomes in A Example 4 If 1 person is selected from the population as described, find the probability that the person a. is male, given that this person is divorced. b. is divorced, given that this person is a man. 3 Q: Is Expected value very useful? A: YEAH! It can be applied to many areas, e.g., to calculate premiums on insurance policies, to weigh the risks versus the benefits of alternatives in business ventures, to indicate to a player of chance what’ll happen if the game is played repeatedly, etc. Example 5 A family with 3 kids can have 0, 1, 2, or 3 boys. Find the expected value for the number of boys in a 3-kid family. Outcomes: Probability No. of boys 0 1 2 3 Example 6 An oil company is considering two sites on which to drill: Site A Site B Profit if oil is found (million USD) 80 120 Loss if no oil is found (million USD) 10 18 Probability of finding oil 0.2 0.1 Which site has the larger expected profit? By how much? 4 Suggested HW 11.7 : # 43, 45, 47, 68, 69 11.8 : # 8 11.7 For #43, 45 and 47 An ice chest contains 6 cans of apple juice, 8 cans of grape juice, 4 cans of orange juice, and 2 cans of mango juice. Suppose 3 cans are selected randomly in succession. Find the probability of selecting #43. THREE cans of apple juice. #45. A can of grape juice, then a can of orange juice, then a can of mango juice. #37. No grape juice. For #68 and #69 Refer to the table in Example 4 from lecture notes. If 1 person is selected from the population as described, find the probability that the person #68. is female, given that this person is divorced. #69. is widowed, given that this person is a woman. 11.8 #8. A 25-year-old can purchase a one-year life insurance policy for $10,000 at a cost of $100. Past history shows that the probability of a person dying at age 25 is 0.002. Find the company’s expected gain per policy. 5

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