4.5: Geometric Probability

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4.5 Geometric Probability

• p. 551-558

GSEs Primary
Primary GSE
M(DSP)105 Solves problems involving
experimental or theoretical probability.
Secondary GSEs
M(GM)102 Makes and defends conjectures,
constructs geometric arguments, uses geometric
properties, or uses theorems to solve problems
involving angles, lines, polygons, circles, or
right triangle ratios (sine, cosine, tangent)
within mathematics or across disciplines or
contexts (e.g., Pythagorean Theorem, Triangle
Inequality Theorem).
M(GM)106 Solves problems involving perimeter,
circumference, or area of two dimensional
figures (including composite figures) or surface
area or volume of three
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• Probability

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Probability

• Definition - a from 0 to 1 that represents the
  chance that an event will occur.
• 0 no chance
• 1 100 chance (the event will always occur).
• .5 or ½ - 50 chance

.5
0
1
Could go either way
No chance
Def. gonna happen
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• Geometric Probability probability
• involving lengths or areas.

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Length Probability Postulate

• If a point on AB is chosen at random and C is
  between A and B, then the probability that the
  point is on AC is Length of AC
• 
  Length of AB

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Example
Find the probability that a point chosen at
random in AF is also part of each of the segments
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Area Problems

• If a point in a region A is chosen at random,
  then the probability that the point is in region
  B, which is in the interior region A, is Area
  of Region B
• Area of Region
  A
• Note. Does not always have to be same shapes.
  Could be a circle inside a square, triangle
  inside a circle, etc. Remember the formulas.

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Example

• A common game is darts. What is the probability
  of randomly throwing a dart such that it hits
  within the red area, given that the dart will
  always land within the boundary of the outer
  circle?
• P(Red)

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1
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Problems

• A dart is thrown at random onto a board that has
  the shape of a circle as shown below.
• Calculate the probability that the dart will hit
  the shaded region. (Use p 3.14 )

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If a dog had an accident in the house, what is
the probability of it occurring in the bedroom ?
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Problem

• The figure shows a circle divided into sectors of
  different colors. If a point is selected at
  random in the circle, calculate the probability
  that it lies
• a) in the red sector.b) in the green sector.c)
  in the blue sector.
• d) in any sector except the green sector.

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Square ABCO contains part of a circle. What is
the probability that a point Chosen at random
would be in the shaded part?
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Problem

• An arrow is shot at random onto the rectangle
  PQRS. Calculate the probability that the arrow
  strikes
• a) triangle AQB.
• b) a shaded region.
• c) either triangle BRC or the unshaded
  region.

• In the figure below, PQRS is a rectangle, and A,
  B, C, D are the midpoints of the respective sides
  as shown.

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Homework