4.5: Geometric Probabilityp. 551-558

M(DSP)–10–5 Solves problems involving experimental or theoretical probability.

GSE’sPrimary

Secondary GSE’sM(G&M)–10–2 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(G&M)–10–6 Solves problems involving perimeter, circumference, or area of two dimensional figures (including composite figures) or surface area or volume of three

Primary GSE

Probability= iespossibilit

happen to what we

total

want

ProbabilityProbability• DefinitionDefinition - a # from 0 to 1 that - a # from 0 to 1 that

represents the chance that an event represents the chance that an event will occur.will occur.

• 0 – no chance0 – no chance

• 1 – 100% chance (the event will always 1 – 100% chance (the event will always occur).occur).

• .5 or ½ - 50% chance.5 or ½ - 50% chance

0 1

No chance Def. gonna happen

.5Could go either way

•Geometric ProbabilityGeometric Probability – probability – probability

involving lengths or areas.involving lengths or areas.

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

Example

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

DF

BD

AE

AD

.4

.3

.2

.1

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.

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)=circle wholeof

circle Red of

area

area

1

5

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 π =3.14 )

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

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.

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

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.

Homework