Solving Probability Problems by Using Combinations

Section 4.9: Solving Probability Problems by Using Combinations This section combines the counting skills we learned in the last two sections with the probability skills we learned earlier in the chapter. This section can be really tricky if you don’t completely understand everything we have done in the chapter so far. If you don’t feel comfortable with something we did earlier in the chapter visit me to get things straight before proceeding. Here is a probability formula that we used earlier in this chapter. The formula isn’t new, but in this section the numbers that we put in the numerator as well as the number we put in the denominator will contain combinations. Here is a basic probability formula that we have already used: 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡ℎ𝑒 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑤𝑖𝑡ℎ 𝑛𝑜 𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛 Example: A club consists of four men and five women. Three members are to be selected at random to form a committee. What is the probability that the committee will consist of three women? (Write your answer as a percent and round to 1 decimal.) The order in which the three members are selected is not important (since we are not attaching labels like President, Vice-President and Secretary) so we may use combinations. P(committee consists of three women) = Number of possible committees with three women Total number of 3 member committees I need 3 of 5 women for the numerator so the numerator will be 3C5. I need any 3 of the 9 club members for the denominator. The denominator will be 9C3. 5C3 9C3 10 = 84 = 11.9% So the probability of the three person committee that is randomly selected from a club of four men and five women being composed of only women is about 11.9 %. Answer: 11.9% Example: 2 cards are drawn from a standard deck of cards. What is the probability they are both red? (Write your answer as a percent and round to 1 decimal.) For numerator I need 2 of the 26 red cards in a deck. The numerator will be 26C2. For the denominator I can get any 2 of the 52 cards in a deck. The denominator will be 52C2. 𝑃(𝑏𝑜𝑡ℎ 𝑐𝑎𝑟𝑑𝑠 𝑎𝑟𝑒 𝑟𝑒𝑑) = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑜𝑢𝑝𝑠 𝑜𝑓 2 𝑜𝑓 26 𝑟𝑒𝑑 𝑐𝑎𝑟𝑑𝑠 26C2 325 = = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑜𝑢𝑝𝑠 𝑜𝑓 2 𝑜𝑓 52 𝑡𝑜𝑡𝑎𝑙 𝑐𝑎𝑟𝑑𝑠 52C2 1326 Answer: about 24.5% Homework #1-16 (Write your answer as a percent rounded to 1 decimal place.) 1) A class consists of 19 girls and 15 boys. If 5 of the students are to be selected at random, determine the probability they are all girls. 2) A club has 8 boys and 5 girls. 3 members are selected at random, determine the probability they are all boys. 3) Of 35 people attending a dance 28 have a college degree. If 4 people at the dance are selected at random, find the probability each has a college degree. 4) Of 50 people attending a concert 30 have seen the group before. If 8 people at the concert are selected at random, find the probability each has seen the group before. 5) A class of 16 people contains 4 people whose birthday is in October. If 3 people from the class are selected at random, find the probability that none of those selected has an October birthday. 6) A class of 25 people contains 5 Gemini’s. If 4 people from the class are selected at random, find the probability that none of those selected are Gemini’s. 7) A committee of 4 is to be randomly selected from a group of 7 teachers and 8 students. Find the probability the committee will consist of 4 students. 8) A committee of 4 is to be randomly selected from a group of 7 teachers and 8 students. Find the probability the committee will consist of 4 teachers 9) Anthony’s wallet contains 8 bills of the following denominations, four $5 bills, two $10 bills, one $20 bill and one $50 bill. If he selects two bills at random, determine the probability that he selects 2 $5 bills. 10) Anthony’s wallet contains 8 bills of the following denominations, four $5 bills, two $10 bills, one $20 bill and one $50 bill. If he selects two bills at random, determine the probability that he selects 2 tens. 11) 3 cards are drawn from a standard deck of cards. What is the probability they are all hearts? 12) 4 cards are drawn from a standard deck of cards. What is the probability they are all clubs? 13) 2 cards are drawn from a standard deck of cards. What is the probability they are both red? 14) 2 cards are drawn from a standard deck of cards. What is the probability they are both black? 15) 2 cards are drawn from a standard deck of cards. What is the probability they are all queens? 16) 2 cards are drawn from a standard deck of cards. What is the probability they are all jacks? Let’s do a more complicated example before getting to the next group of problems. Example: A class consists of 10 girls and 9 boys. If 5 of the students are to be selected at random, determine the probability that 3 boys and 2 girls are selected. For the numerator, I need to select: 3 of 9 boys: This will give a 9C3 in the numerator. 2 of 10 girls: This will give a 10C2 in the numerator. For the denominator, I can select: 5 of 19 club members: The denominator will be 19C5. 𝑃(3 𝑏𝑜𝑦𝑠 𝑎𝑛𝑑 2 𝑔𝑖𝑟𝑙𝑠 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑) = = 9𝐶3∗10𝐶2 19𝐶5 84∗45 = 11628 = .3250 Answer: 32.5% (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑜𝑢𝑝𝑠 𝑤𝑖𝑡ℎ 3 𝑜𝑓 9 𝑏𝑜𝑦𝑠)(𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑜𝑢𝑝𝑠 𝑤𝑖𝑡ℎ 2 𝑜𝑓 10 𝑔𝑖𝑟𝑙𝑠) 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑜𝑢𝑝𝑠 𝑜𝑓 5 𝑜𝑓 19 𝑐𝑙𝑢𝑏 𝑚𝑒𝑚𝑏𝑒𝑟𝑠) Homework #17-32 (Write your answer as a percent rounded to 1 decimal place.) 17) A class consists of 19 girls and 15 boys. If 12 of the students are to be selected at random, determine the probability that 4 boys and 8 girls are selected. 18) A club has 8 boys and 5 girls. 3 members are selected at random, determine the probability that 2 boys and 1 girl is selected. 19) Of 35 people attending a dance 28 have a college degree. If 4 people at the dance are selected at random, find the probability 3 have a college degree and 1 does not. 20) Of 50 people attending a concert 30 have seen the group before. If 8 people at the concert are selected at random, find the probability that 6 have seen the group before and 2 have not. 21) A class of 16 people contains 4 people whose birthday is in October. If 3 people from the class are selected at random, find the probability that 2 have a birthday in October and 1 doesn’t. 22) A class of 25 people contains 5 Gemini’s. If 4 people from the class are selected at random, find the probability that 1 Gemini and 3 people that are not Gemini’s are selected. 23) A committee of 4 is to be randomly selected from a group of 7 teachers and 8 students. Find the probability the committee will consist of 3 students and 1 teacher. 24) A committee of 6 is to be randomly selected from a group of 7 teachers and 8 students. Find the probability the committee will consist of 4 teachers and 2 students. 25) Anthony’s wallet contains 11 bills of the following denominations, four $5 bills, five $10 bills, one $20 bill and one $50 bill. If he selects five bills at random, determine the probability that he selects 2 $5 bills and 3 tens. 26) Anthony’s wallet contains 10 bills of the following denominations, four $5 bills, two $10 bills, three $20 bill and one $50 bill. If he selects 3 bills at random, determine the probability that he selects 2 tens and a five. 27) 3 cards are drawn from a standard deck of cards. What is the probability they are two hearts and 1 spade? 28) 4 cards are drawn from a standard deck of cards. What is the probability they are 2 clubs and 2 diamonds? 29) 2 cards are drawn from a standard deck of cards. What is the probability they are 1 red and 1 black? 30) 5 cards are drawn from a standard deck of cards. What is the probability they are 3 black and 2 red? Answers: 1) 19C5 / 34C5 = .0417 4.2% 3) 28C4 / 35C4 = .391 = 39.1% 5) 4C3 / 16C3 = .007 = 0.7% 7) 8C4 / 15C4 = .051 = 5.1% 9) 4C2 / 8C2 = .214 = 21.4% 11) 13C3 / 52C3 =.013 = 1.3% 13) 26C2 / 52C2 = .245 = 24.5% 15) 4C2 / 52C2 = .0045 = .5% 17) (15C4 * 19C8) / 34C12 = .188 = 18.8% 19) (28C3 * 7C1)/35C4 = .438 = 43.8% 21) (4C2 * 12C1) / 16C3 = .129 = 12.9% 23) (8C3 * 7C1) / 15C4 = .287 = 28.7% 25) (4C2 * 5C3) / 11C5 = .1298 = 13% 27) (13C2 * 13C1) / (52C3) = .046 = 4.6% 29) (26C1 * 26C1) / 52C2 = .5098 = 51%