Geometry (B): READ: Geometric Probability Reading

Geometric Probability

Learning Objectives

Identify favorable outcomes and total outcomes.
Express geometric situations in probability terms.
Interpret probabilities in terms of lengths and areas.

Introduction

You’ve probably studied probability before now (pun intended). We’ll start this lesson by reviewing the basic concepts of probability.

Once we’ve reviewed the basic ideas of probability, we’ll extend them to situations that are represented in geometric settings. We focus on probabilities that can be calculated based on lengths and areas. The formulas you learned in earlier lessons will be very useful in figuring these geometric probabilities.

Basic Probability

Probability is a way to assign specific numbers to how likely, or unlikely, an event is. We need to know two things:

the total number of possible outcomes for an event. Let’s call this $t$ .
the number of “favorable” outcomes for the event. Let’s call this $f$ .

The probability of the event, call it $P$ , is the ratio of the number of favorable outcomes to the total number of outcomes.

Definition of Probability

$P=\frac{f}{t}$

Example 1

Nabeel’s company has $12$ holidays each year. Holidays are always on weekdays (not weekends). This year there are $260$ weekdays. What is the probability that any weekday is a holiday?

There are $260$ weekdays in all.

$t=260$

$12$ of the weekdays are holidays

$f & =12 \\ P & =\frac{f}{t}=\frac{12}{260} \approx 0.05$

Comments: Probabilities are often expressed as fractions, decimals, and percents. Nabeel can say that there is a $5\%$ chance of any weekday being a holiday. Note that this is (unfortunately?) a relatively low probability.

Example 2

Charmane has four coins in a jar: two nickels, a dime, and a quarter. She mixes them well. Charmane takes out two of the coins without looking. What is the probability that the coins she takes have a total value of more than $\$0.25$ ?

$t$ in this problem is the total number of two-coin combinations. We can just list them all. To make it easy to keep track, use these codes: $N1$ (one of the nickels), $N2$ (the other nickel), $D$ (the dime), and $Q$ (the quarter).

Two-coin combinations:

$N1,N2 && N1,D && N1,Q && N2,D && N2,Q && D, Q$

There are six two-coin combinations.

$t = 6$

Of the six two-coin combinations, three have a total value of more than $\$0.25$ . They are:

$N1,Q (\$0.30) && N2,Q (\$0.30) && D,Q (\$0.35)$

$f = 3$

The probability that the two coins will have a total value of more than $\$0.25$ is $P=\frac{f}{t}=\frac{3}{6}=\frac{1}{2}=0.5=50\%$ .

The probability is usually written as $\frac{1}{2},0.5$ , or $50\%$ . Sometimes this is expressed as “a $50-50$ chance” because the probability of success and of failure are both $50\%$ .

Geometric Probability

The values of $t$ and $f$ that determine a probability can be lengths and areas.

Example 3

Sean needs to drill a hole in a wall that is $14 \;\mathrm{feet}$ wide and $8 \;\mathrm{feet}$ high. There is a $2-\mathrm{foot}-$ by $-3-\mathrm{foot}$ rectangular mirror on the other side of the wall so that Sean can’t see the mirror. If Sean drills at a random location on the wall, what is the probability that he will hit the mirror?

The area of the wall is $14\times 8=112 \;\mathrm{square\ feet}$ . This is $t$ .

The area of the mirror is $2\times 3=6 \;\mathrm{square\ feet}$ . This is $f$ .

The probability is $P=\frac{6}{112} \approx 0.05$ .

Example 4

Ella repairs an electric power line that runs from Acton to Dayton through Barton and Canton. The distances in miles between these towns are as follows.

$\mathrm{Barton\ to\ Canton} = 8\;\mathrm{miles}.$
$\mathrm{Acton\ to\ Canton} = 12\;\mathrm{miles}.$
$\mathrm{Canton\ to\ Dayton} = 2\;\mathrm{miles}.$

If a break in the power line happens, what is the probability that the break is between Barton and Dayton?

Approximately $71\%$ .

$t & = \text{the distance from Acton to Dayton} = 4 + 8 + 2 = 14\ \text{miles}. \\ f & = \text{the distance from Barton to Dayton} = 8 + 2 = 10\ \text{miles}. \\ P & =\frac{f}{t}=\frac{10}{14}=\frac{5}{7} \approx 0.71=71\%$

Lesson Summary

Probability is a way to measure how likely or unlikely an event is. In this section we saw how to use lengths and areas as models for probability questions. The basic probability ideas are the same as in non-geometry applications, with probability defined as:

$\text{Probability}=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$

Points to Consider

Some events are more likely, and some are less likely. No event has a negative probability! Can you think of an event with an extremely low, or an extremely high, probability? What are the ultimate extremes—the greatest and the least values possible for a probability? In ordinary language these are called “impossible” (least possible probability) and “certain” or a “sure thing” (greatest possible probability).

The study of probability originated in the seventeenth century as mathematicians analyzed games of chance.

For Further Reading

French mathematicians Pierre de Fermat and Blaise Pascal are credited as the “inventors” of mathematical probability. The reference below is an easy introduction to their ideas. http://mathforum.org/isaac/problems/prob1.html

The following questions are for your own review. The answers below are for you to check your work and understanding.

Review Questions

Rita is retired. For her, every day is a holiday. What is the probability that tomorrow is a holiday for Rita?
Chaz is “on call” any time, any day. He never has a holiday. What is the probability that tomorrow is a holiday for Chaz?
The only things on Ray’s refrigerator door are green magnets and yellow magnets. Ray takes one magnet off without looking.
1. What is the probability that the magnet is green?
2. What is the probability that the magnet is yellow?
3. What is the probability that the magnet is purple?
Ray takes off two magnets without looking.
1. What is the probability that both magnets are green?
2. What is the probability that Ray takes off one green and one yellow magnet?
Reed uses the diagram below as a model of a highway.
1. What is the probability that the accident is not between Canton and Dayton?
2. What is the probability that the accident is closer to Canton than it is to Barton?

Reed got a call about an accident at an unknown location between Acton and Dayton.

A tire has an outer diameter of $26\;\mathrm{inches}$ . Nina noticed a weak spot on the tire. She marked the weak spot with chalk. The chalk mark is $4\;\mathrm{inches}$ along the outer edge of the tire. What is the probability that part of the weak spot is in contact with the ground at any time?
Mike set up a rectangular landing zone that measures $200\;\mathrm{feet}$ by $500\;\mathrm{feet}$ . He marked a circular helicopter pad that measured $50$ feet across at its widest in the landing zone. As a test, Mike dropped a package that landed in the landing zone. What is the probability that the package landed outside the helicopter pad?
Fareed made a target for a game. The target is a $4-$ foot-by $-4$ -foot square. To win a player must hit a smaller square in the center of the target. If the probability that players who hit the target win is $20\%$ , what is the length of a side of the smaller square?
Amazonia set off on a quest. She followed the paths shown by the arrows in the map.

Every time a path splits, Amazonia takes a new path at random. What is the probability that she ends up in the cave?

Review Answers

$1, 100\%$ , or equivalent
$0$
1. $\frac{2}{5}, 0.4, 40\%$ , or equivalent
2. $\frac{3}{5}, 0.6, 60\%$ , or equivalent
3. $0$
4. $\frac{2}{15} \approx 0.13$ or equivalent
5. $\frac{4}{15} \approx 0.27$ or equivalent
1. $\frac{12}{14}=\frac{6}{7} \approx 0.86=86\%$
2. $\frac{6}{14}=\frac{3}{7} \approx 0.43=43\%$
Approximately $0.05=5\%$
Approximately $98\%$
Approximately $1.78\;\mathrm{feet}$
$\frac{5}{12} \approx 0.42$ or equivalent

Last modified: Tuesday, June 29, 2010, 12:01 PM