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

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 = /20 = 3/10

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

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

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

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.)

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?

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%

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?

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 ) = ____________ /60 = 3/4 ¾ or 75 %

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.

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

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

Mrs. McConaughyGeoemtric Probability 2/9

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?

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

Mrs. McConaughyGeoemtric Probability Homework Assignment: