Download Chapter Thirteen: Probability

Chapter Thirteen: Probability
Section One: Theoretical Probability
Theoretical probability is what we think will happen if an event did take place. Weather
prediction is an example of theoretical probability. All the things that could happen is
called the sample space.
EX1: Find the sample space for the following experiments.
a. rolling two die
b. tossing three coins
The favorable outcome is the one that we are looking to happen. It’s the one we are trying
to find the chances of. The formula for finding the theoretical probability is
P( E ) 
number of favorable outcomes
number of elements in sample space
EX2: A dice is rolled twice. The sample space is shown below. Find the probability of
each event.
1,1 1, 2  1,3 1, 4  1,5  1,6 
 2,1  2, 2   2,3  2, 4   2,5   2,6 
 3,1  3, 2   3,3  3, 4   3,5   3,6 
 4,1  4, 2   4,3  4, 4   4,5   4,6 
 5,1  5, 2   5,3  5, 4   5,5   5,6 
 6,1  6, 2   6,3  6, 4   6,5   6,6 
a. P  sum of 9
b. P  sum of 12
c. P  sum  6
d. P  sum  6
EX3: On the first day of class, a teacher puts a sticker under 19 out of 23 chairs in her
classroom. A second teacher puts a sticker under 15 out of 20 chairs in her classroom.
Students who sit in the chairs with stickers will receive a $5 gift certificate to the school
cafeteria. In which classroom is the probability of receiving a gift certificate greater?
Section Two: Counting the Elements of a Set
Two events that can occur at the same time are called inclusive events. If they cannot we
refer to them as mutually exclusive.
EX1: Tell whether the following events are inclusive or mutually exclusive.
a. Drawing an 8 from a deck of cards and drawing a heart
b. Rolling an odd number on a dice and rolling a 4
c. Picking a number that is an even number and choosing a multiple of 3
We have to be careful when finding the probability of combined events. We have to take
into consideration whether the events are inclusive or mutually exclusive.
EX2: How many cards in a standard deck of 52 playing cards hearts OR kings? Draw a
Venn diagram to illustrate.
EX3: A bag contains 3 blue marbles, 4 red marbles, and 5 green marbles. You reach into
the bag and randomly draw a marble. Find the following probabilities.
a. P (blue)
b. P (red)
c. P(blue or red)
d. P (green)
e. P(blue or green)
EX4: If you draw a card at random from a complete deck of 52 playing cards, what is the
probability that you will draw a spade or a club?
EX5: What is the probability of drawing a red face card OR a heart from a standard deck
of 52 playing cards?
EX6: What is the probability of drawing an odd-numbered spade OR an ace from a deck
of playing cards?
Section Three: The Fundamental Counting Principle
Sometimes we need to determine how many different ways that a series of events can
happen. There are two different methods for doing this. One method is by using a tree
diagram. However, if there are multiple events then we can use the fundamental counting
principle.
EX1: A restaurant serves four kinds of sandwiches: turkey, roast beef, chicken, or
vegetarian. With each sandwich, the restaurant offers a free beverage: coffee, tea, or milk.
Use a tree diagram to find the number of sandwich-and-beverage combinations that the
restaurant offers.
The other method for working this problem is with the fundamental counting principle:
1. Draw a blank for each event happening
2. How many choices do you have for each event
3. Multiply the numbers
EX2: Work example two using the fundamental counting principle.
EX3: A certain style T-shirt is manufactured in 4 sizes. Each size is available in 9
different colors. How many different size-and-color combinations must a clothing store
stock in order to carry the complete line of these T-shirts?
EX4: A print shop offers custom-made personalized stationary. The customer can choose
white, pink, or blue paper. A border around the edge can be either classic or modern in
style. The border can be printed in either black, blue, brown, red, purple, or green ink.
How many choices of stationary are possible?
EX5: Two standard decks of cards are shuffled separately. Then a card is drawn from
each deck. In how many ways can you draw an ace from the first deck and a heart from
the second deck?
Section Four: Independent Events
Two events are independent if the occurrence of one event does not affect the probability
of the occurrence of the other event. If the events were dependent, the occurrence of the
first will impact the likelihood of the second.
Independent Events
1. Rolling an even number on a dice and flipping a heads on a coin
2. Spinning a spinner and then pulling a marble from a bag
3. Pulling a red marble from a bag, replacing it, and pulling a blue marble out
Dependent Events
1. Pulling a blue marble from a bag, not replacing it, and then pulling out a red
marble
2. Choosing a date from a calendar and then choosing a different date
We use a method similar to the Fundamental Counting Principle when find the
probability of multiple events occurring. Use a blank for each separate event and put the
probability of each in the blank.
EX1: Two fair die, one red and one green, are rolled at the same time. What is the
probability that the red dice shows an even number and the green dice shows a prime
number?
EX2: In a softball game, Cystal is at bat, and Karen is next at bat. Crystal gets on base in
65% of her at bats. Karen gets on base in 40% of her at bats. What is the probability that
both girls will get on base? What is the probability that neither will get on base? What is
the probability that Crystal gets on base and Karen does not?
EX3: Two dice are rolled. Find the probability that both cubes show a number less than 3.
Sometimes it is easier to find the chances that something will not happen and use that
probability to find the chance that it will happen. These two cases together are called
complements. An event and its complement together will have a probability of one.
EX4: Two fair die are rolled. Find the probability that at least one dice shows a number
greater than 2.
EX5: If two die are rolled, what is the probability that the sum is greater than or equal
to 3?
Section Five: Simulations
Sometimes instead of actually performing an experiment to find the chances of something
happening, we can set up a similar mock experiment called a simulation. Many times we
will use a set of random numbers to simulate an experiment. Our graphing calculators can
generate these numbers that are equally likely. We will perform several trials to find the
probability. The more trials we use, the more accurate our probability will be.
EX1: Suppose that a weather report gives a 60% chance of rain on Satuday and a 60%
chance of rain on Sunday. Create a simulation, perform 25 trials, and display the results.
According to the results, what is the probability that it will rain on both days.
EX2: Suppose that 90% of the flights for an airline arrive on time. Create a simulation,
perform 25 trials, and display the results. According to the results, what is the probability
that three consecutive flights will arrive on time?