1. Mrs. McConaughyGeoemtric Probability Geometric Probability During this lesson, you will determine probabilities by calculating a ratio of two lengths, areas, or volumes.

2. Mrs. McConaughyGeoemtric Probability PROBABILITIES If you roll a 20-sided die with numbers 1-20, what is the probability of rolling a number divisible by 3? Favorable outcomes: _____ Total possible Outcomes: _____ P(event) = favorable outcomes total possible outcomes P(event) = ____________ 3, 6, 9, 12, 15, 18 = 6 6 20 6/20 = 3/10

3. Mrs. McConaughyGeoemtric Probability GEOMETRIC PROBABILITY Some probabilities are found by calculating a ratio of two lengths, areas, or volumes. Such probabilities are called _______________________. geometric probabilities

4. Mrs. McConaughyGeoemtric Probability EXAMPLE: A gnat lands at a random point on the ruler’s edge. Find the probability that the point is between 3 and 7. (Assume that the ruler is 12 inches long.) P(landing between 3 and 7) = length of favorable segment length of whole segment 4/12 = 1/3

5. Mrs. McConaughyGeoemtric Probability CHECK: A point on AB is selected at random. What is the probability that it is a point on CD? P(event) = ________________ A C D B Length of CD Length of BD = 4/12 = 1/3

6. Mrs. McConaughyGeoemtric Probability EXAMPLE: GEOMETRIC PROBABILITY A gnat lands at a random point on the edge of the ruler below. Find the probability that the point is between 2 and 10. (Assume that the ruler is 12 inches long.)

7. Mrs. McConaughyGeoemtric Probability COMMUTING: D. A. Tripper’s bus runs every 25 minutes. If he arrives at his bus stop at a random time, what is the probability that he will have to wait at least 10 minutes for the bus?

8. Mrs. McConaughyGeoemtric Probability If D.A. Tripper arrives at any time between A and C, he has to wait at least 10 minutes until B. P(waiting at least 10 minutes) = _____________ What is the probability that D.A. Tripper will have to wait more than 10 minutes for the bus? __________ Solution: Solution: Assume that a stop takes very little time, and let AB represent the 25 minutes between buses. A C B 3/5 or 60% 2/5 or 40%

9. Mrs. McConaughyGeoemtric Probability EXAMPLE 2: A museum offers a tour every hour. If Dino Sur arrives at the tour site at a random time, what is the probability that he will have to wait for at least 15 minutes?

10. Mrs. McConaughyGeoemtric Probability Solution: Because the favorable time is given in minutes, write 1 hour as 60 minutes. Dino may have to wait anywhere between 0 minutes and 60 minutes. Represent this using a segment: Starting at 60 minutes, go back 15 minutes. The segment of length _____ represents Dino’s waiting more than 15 minutes. P (waiting more than 15 minutes) = ____________. P( waiting at least 15 minutes ) = ____________. 45 45/60 = 3/4 ¾ or 75 %

11. Mrs. McConaughyGeoemtric Probability EXAMPLE 4: A square dartboard is represented in the accompanying diagram. The entire dartboard is the first quadrant from x = 0 to 6 and from y = 0 to 6. A triangular region on the dartboard is enclosed by the graphs of the equations y = 2, x = 6, and y = x. Find the probability that a dart that randomly hits the dartboard will land in the triangular region formed by the three lines.

12. Mrs. McConaughyGeoemtric Probability Solution: The first step is to graph the three lines that are given and determine the area of the triangle. The formula for the area of a triangle is _________, and the base of the triangle is ______ and the height is ______. Through substitution, the area of our triangle is found to be ______. Hitting this area with the dart is the desired event and the number 8 will be the numerator of our probability fraction. Area ∆ = ½ bh 4 units 8 units

13. Mrs. McConaughyGeoemtric Probability Solution: We could reduce our fraction or convert it to a decimal or percent, but these additional steps are not necessary. The probability of a dart that randomly hits the dartboard landing in the triangular region is _____, or _____. 2/98/36

14. Mrs. McConaughyGeoemtric Probability 2/9

15. Mrs. McConaughyGeoemtric Probability Final Checks for Understanding Express elevators to the top of the PPG Place leave the ground floor every 40 seconds. What is the probability that a person would have to wait more than 30 seconds for an express elevator?

16. Mrs. McConaughyGeoemtric Probability You throw a dart at the board shown. Your dart is equally likely to hit any point inside the square board. Are you more likely to get 10 points or 0 points? Final Checks for Understanding

17. Mrs. McConaughyGeoemtric Probability Homework Assignment: