Question 10 - End-of-Chapter Exercises - Chapter 7 - The Mathematics of Maybe: Introduction to Probability

Transcript

Question 10 A basket contains 4 red balls and 5 blue balls. One ball is drawn and laid aside, and a second ball is drawn. Draw a tree diagram to represent the possible outcomes and probabilities. Use the tree diagram to answer the following questions. (i) What is the probability of drawing a red ball and then a blue ball? (ii) What is the probability of drawing 2 blue balls? Here, the ball is taken out without replacement So, our probabilities when picking second ball needs to be carefully calculated Now, let’s answer our question (i) What is the probability of drawing a red ball and then a blue ball? Now, Probability of picking red ball first = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑒𝑑 𝑏𝑎𝑙𝑙𝑠)/(𝑇𝑜𝑡𝑎𝑙 𝑏𝑎𝑙𝑙𝑠) = 𝟒/𝟗 Now, after red ball is picked, we have 8 balls left, out of which 5 are blue Probability of picking blue ball after red = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑙𝑢𝑒 𝑏𝑎𝑙𝑙𝑠 𝑙𝑒𝑓𝑡)/(𝑇𝑜𝑡𝑎𝑙 𝑏𝑎𝑙𝑙𝑠 𝑙𝑒𝑓𝑡) = 𝟓/𝟖 Thus, Probability of drawing a red ball and then a blue ball = 𝟒/𝟗 × 𝟓/𝟖 = 20/72 = 𝟓/𝟏𝟖 (ii) What is the probability of drawing 2 blue balls? Here, we first pick blue ball, and then pick blue ball again Now, Probability of picking blue ball first = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑙𝑢𝑒 𝑏𝑎𝑙𝑙𝑠)/(𝑇𝑜𝑡𝑎𝑙 𝑏𝑎𝑙𝑙𝑠) = 𝟓/𝟗 Now, after blue ball is picked, we have 8 balls left, out of which 3 are blue Probability of picking blue ball after blue = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑙𝑢𝑒 𝑏𝑎𝑙𝑙𝑠 𝑙𝑒𝑓𝑡)/(𝑇𝑜𝑡𝑎𝑙 𝑏𝑎𝑙𝑙𝑠 𝑙𝑒𝑓𝑡) = 𝟒/𝟖 Thus, Probability of drawing two blue balls = 𝟓/𝟗 × 𝟒/𝟖 = 20/72 = 𝟓/𝟏𝟖 Thus, Probability of drawing a red ball and then a blue ball = 𝟒/𝟗 × 𝟓/𝟖 = 20/72 = 𝟓/𝟏𝟖 (ii) What is the probability of drawing 2 blue balls? Here, we first pick blue ball, and then pick blue ball again Now, Probability of picking blue ball first = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑙𝑢𝑒 𝑏𝑎𝑙𝑙𝑠)/(𝑇𝑜𝑡𝑎𝑙 𝑏𝑎𝑙𝑙𝑠) = 𝟓/𝟗 Now, after blue ball is picked, we have 8 balls left, out of which 3 are blue Probability of picking blue ball after blue = (𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑙𝑢𝑒 𝑏𝑎𝑙𝑙𝑠 𝑙𝑒𝑓𝑡)/(𝑇𝑜𝑡𝑎𝑙 𝑏𝑎𝑙𝑙𝑠 𝑙𝑒𝑓𝑡) = 𝟒/𝟖 Thus, Probability of drawing two blue balls = 𝟓/𝟗 × 𝟒/𝟖 = 20/72 = 𝟓/𝟏𝟖