mrs. mcconaughygeoemtric probability geometric probability during this lesson, you will determine...
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Mrs. McConaughy Geoemtric Probability
Geometric Probability
During this lesson, you will determine probabilities by
calculating a ratio of two lengths, areas, or volumes.
Mrs. McConaughy Geoemtric 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
Mrs. McConaughy Geoemtric Probability
GEOMETRIC PROBABILITY
Some probabilities are found by calculating a ratio of two lengths, areas, or volumes. Such probabilities are called _______________________. geometric probabilities
Mrs. McConaughy Geoemtric 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. McConaughy Geoemtric 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. McConaughy Geoemtric 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. McConaughy Geoemtric 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. McConaughy Geoemtric 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. McConaughy Geoemtric 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. McConaughy Geoemtric 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 = 3/4¾ or 75 %
Mrs. McConaughy Geoemtric 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. McConaughy Geoemtric 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 units4 units
8 units
y=x
y=2
x=6
14 4 8
2A
Mrs. McConaughy Geoemtric 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 _____.
2/9 8/36
Mrs. McConaughy Geoemtric Probability
y=x
y=2
x=6
14 4 8
2A
event
number of favorable outcomesP _______
number of possible outcomes 2/9
Mrs. McConaughy Geoemtric 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. McConaughy Geoemtric 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
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