Transcript Geometric 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
6
outcomes: _____
P(event) = favorable outcomes
total possible outcomes
3, 6, 9, 12, 15, 18 = 6
6/20 = 3/10
P(event) = ____________
Total possible
Outcomes: _____
20
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?
Length of CD
= 4/12 = 1/3
Length of BD
P(event) = ________________
A
Mrs. McConaughy
C
D
Geoemtric Probability
B
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
Solution:
Solution: Assume that a stop takes very
little time, and let AB represent the 25
minutes between buses.
A
C
Mrs. McConaughy
If D.A. Tripper arrives at any
time between A and C, he has
B
to wait at least 10 minutes
until B.
P(waiting at least 10 minutes)
3/5 or 60%
= _____________
What is the probability that
D.A. Tripper will have to wait
more than 10 minutes for the
2/5 or 40%
bus? __________
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
45 represents Dino’s waiting
segment of length _____
more than 15 minutes.
= 3/4
P (waiting more than 15 minutes) = 45/60
____________.
P( waiting at least 15 minutes ) = ¾
____________.
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:
y=x
y=2
A
1
 4  4   8
2
Mrs. McConaughy
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
Area
∆ = ½ bh and the base of
_________,
the triangle is 4______
units and the
x=6
4 units Through
height is ______.
substitution, the area of our
triangle is found to be
8
units Hitting this area
______.
with the dart is the desired
event and the number 8 will be
the numerator of our
probability fraction.
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
8/36 or _____.
2/9
region is _____,
Mrs. McConaughy
Geoemtric Probability
P event 
number of favorable outcomes 2/9

 _______
number of possible outcomes
y=x
x=6
y=2
Mrs. McConaughy
1
A   4 
4   8Probability
Geoemtric
2
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
Final Checks for Understanding
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?
Mrs. McConaughy
Geoemtric Probability
Homework Assignment:
Mrs. McConaughy
Geoemtric Probability