mrs. mcconaughygeoemtric probability geometric probability during this lesson, you will determine...

Mrs. McConaughy Geoemtric Probability

Geometric Probability

During this lesson, you will determine probabilities by

calculating a ratio of two lengths, areas, or volumes.


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


GEOMETRIC PROBABILITY

Some probabilities are found by calculating a ratio of two lengths, areas, or volumes. Such probabilities are called _______________________. geometric probabilities


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


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


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.)


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?


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%


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?


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 = 3/4¾ or 75 %


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.


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 units4 units

8 units

y=x

y=2

x=6

14 4 8

2A


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 _____.

2/9 8/36


y=x

y=2

x=6

14 4 8

2A

event

number of favorable outcomesP _______

number of possible outcomes 2/9


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?


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


Homework Assignment:

mrs. mcconaughygeoemtric probability geometric probability during this lesson, you will determine...

Documents