5/15/2015 Chapter 11.1 Geometric Probability Goals: Find a geometric probability. Use geometric probability to solve real-life problems. Probability • A probability is a number from 0 to 1 that represents the chance that an event will occur. • Sometimes we say “percent probability” which changes the number to a percentage. This can be helpful to visualize. • A geometric probability is a probability that involves a geometric measure such as length or area. 1 5/15/2015 The Coin vs. the lottery • Lets think about what probability mean. This is the desired outcomes of a situation over the total number of possible outcomes. • 𝐷𝑜 𝑃𝑜 = p desired outcome Lets think about coins: Dice • What is the possible outcome of a die? • What is the possible outcome of 2 dice? • What is the possible outcome of 2 dice faces added together? • All of these are used in probability. 2 5/15/2015 Questions • What is the probability that the die will roll an even number? • What is the probability that the dice will roll a 4 and a 5? • What is the probability that the 2 dice will add up to 7? 11? Geometric probability • Probability and Length: Let 𝐴𝐵be a segment that contains the segment 𝐶𝐷. If a point K on is chosen at random, then the probability that it is on 𝐶𝐷 is as follows: 𝐶𝐷 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓𝐶𝐷 • 𝑃 𝑃𝑜𝑖𝑛𝑡 𝐾 𝑖𝑠 𝑜𝑛 𝐶𝐷 = 𝐴𝐵 = 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐴𝐵 • What does this look like? 3 5/15/2015 Problem • Find the probability that a point chosen at random on 𝐴𝐵 is on 𝐶𝐷: Problems • find the probability that a point A, selected randomly on 𝑨𝑩 is on the given segment. a) 𝐶𝐹 b)𝐸𝐹 c) 𝐶𝐸 4 5/15/2015 The other geometric probability • Here we are looking at the probability of a randomly chosen point falls in a certain area of a diagram. 𝑃 𝑃𝑜𝑖𝑛𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑠ℎ𝑎𝑑𝑒𝑑 𝑎𝑟𝑒𝑎 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑠ℎ𝑎𝑑𝑒𝑑 𝑟𝑒𝑔𝑖𝑜𝑛 𝑡𝑜𝑡𝑎𝑙 𝑎𝑟𝑒𝑎 Example • Find the probability that a point chosen at random in parallelogram ABCD lies in the shaded region. 𝑃 𝑃𝑜𝑖𝑛𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑠ℎ𝑎𝑑𝑒𝑑 𝑎𝑟𝑒𝑎 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑠ℎ𝑎𝑑𝑒𝑑 𝑟𝑒𝑔𝑖𝑜𝑛 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑃𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 5 5/15/2015 You try • Find the probability that a point chosen at random in the figure lies in the shaded region. Real applications • Archery Target Determine the probability for each outcome on the archery target shown. The center ring has a radius of 1 unit. Each successive ring has a radius 1 unit greater than the previous one. Assume the arrow is equally likely to hit any point on the target. Get into groups of 2 or three and solve the questions below. a. 25 points b. 20 points c. 15 points d. 10 points 6 5/15/2015 Real-Life Application • Carnival Game: A game at a local carnival involves tossing beanbags at the target shown below. A person wins a prize if the beanbag goes through the eyes, nose, or part of the mouth. In a group of 3, answer the following questions rounding to 3 decimals. 1. Find the probability that someone throws the beanbag through the left eye. 2. Find the probability that someone throws the beanbag through either eye. 3. Find the probability that someone throws the beanbag through the mouth. 4. Find the probability that someone will win a prize. 7