1. Probability three is 1/6. If necessary, you may need to
simplify your fraction, but in this case the
fraction is already in simplest form.
Probability is a fraction where the Probability can be addressed, however, in
numerator is the number of times the event a percent or a decimal along as well as
occurs and the denominator is the total number fractions. To determine the probability in a
of trials. It is represented by number between 0 decimal, you must first determine what the
and 1 (or 0% to 100%), and it is used to show fraction probability is, which is 1/6 in this
the likelihood of an event’s occurrence. In example. Then, you divide the numerator by
order to find probability, you must first count the denominator, and your quotient should be a
how many trials there are and how many times decimal value, which is your decimal. If your
the event occurs. If it doesn’t tell you how decimal is repeating or is very long, you may
many trials there are, you may need to add up want to consider rounding it to the nearest
all of the events that occurred assuming it gives hundredth.
you that data. The event that occurs is the thing It is possible to find a percent probability
that you want to happen, for example rolling a from both a decimal and a fraction. To
three on a fair number cube. The amount of determine a percent from a decimal, all you
trials there are is the amount of outcomes that need to do is move the decimal 2 digits to the
can occur. When you roll a fair number cube, right. For example, with .200, you would move
the possible outcomes that can occur are the decimal place twice to the right, which gets
1,2,3,4,5, and 6, a total of 6 possible you 20.0, 20.0%, or 20%. To determine a
outcomes. Since there are six possible probability from a fraction, you must set up a
outcomes (denominator) and one event that can proportion, one of your ratios being your
occur (numerator), the probability of rolling a fraction that you already know and the other

2. ratio being x over 100 (because a percent is a probability that she will pick a nine in a
fraction where the denominator is assumed as percent?
being 100). Then, you must use the cross
product method where you multiply the first Out of the 52 to cards, or the total number of
ratio’s denominator by the second ratio’s trials, 4 events could occur, the four “9” cards
numerator and the first ratio’s numerator by the in the deck (nine of hearts, diamonds, spades,
second ratio’s denominator. One of your and clubs). The number of times the event can
products out of these two values will be a occur is 4 (numerator), and the number of trials
constant, but one will have a variable in it. or possible outcomes that could happen is 52
These two values are equal, so you should be (denominator) because there are 52 different
able to make an equation that states c = cx cards Juanita could pick from the deck. We
(where c stands for constant and x stands for now have the fraction 4/52. We must now set
variable). You can solve this equation by doing up a proportion indicating how many times the
a one-step, where you will undo the all of the event would have occurred if there were 100
values that restrict the variable to be isolated by trials to get a percent. Your two proportional
doing the inverse operation. Then, to keep an ratios will be 4/52 and x/100. You will first
equal balance, you must perform the same cross-multiply the numerator of the first ratio by
action to the other side (the two sides are the denominator of the second ratio, and those
separated by the equal sign). You should now two numbers are 4 and 100, which equals 400.
get answer that reads c = x (where c is a The other numbers you multiply are 52 and x,
constant and x is a variable. Your constant which equals 52x. The value 52x is equal to
value is your percent. the value 400, and we can represent this
Problem 1: Juanita randomly picks a card from through the equation 400 = 52x. We want to
a standard 52-card deck. What is the

3. isolate x, so we will undo the operations that Experimental probability is determined by
are restricting x from being isolated (that conducting an experiment repeatedly and
operation in this case is *52) by performing the observing the number of times the event occurs in
inverse operation, which in this case is to divide the experiment compared to the total number of
trials. For example, if you rolled a fair number
by 52. You now have isolated x, however you
cube 5 times, and 3 out of the 5 times you rolled a
must now divide 400 by 52 to keep both sides
2, your experimental probability of rolling a 2
of the equation equal. You now have the
would be 3/5. Experimental probability is different
answer, 7.692307 repeating, which can be from theoretical probability because it is based off
rounded to the nearest hundredth as 7.7, or of the outcome of an experiment that was done a
7.7%. limited number of times. The theoretical probability
on the other hand measures the probability of an
Problem 2: John, Chase, Micah, Willie, and Ms. event occurring assuming the trial was performed
Howson competed in a running race. All of the infinity times. In other words, theoretical
5 runners have an equal chance of winning probability shows the likelihood of an event
except for Ms. Howson, who has twice as much occurring IF you performed an experiment, while
of a chance as winning. What is the probability experimental probabilities shows how often the
that Ms. Howson would not win the race? event occurred in an experiment THAT WAS
ALREADY CONDUCTED compared to the total
number of trials.
4/6 or 2/3
Problem 1: D’Andrus flipped a penny 16 times, and
10 out of those 16 times she got “heads”. What is
Experimental Probability the experimental probability that D’Andrus flipped
a “tail”?

4. The experimental probability that D’Andrus flipped
a head was 10/16, because in this experiment, he Theoretical Probability
received the event heads 10 out of the 16 trials.
The best way to find the number of times the event Theoretical probability is found by
did not occur is to subtract the numerator from the dividing the number of times an event could
denominator using the fraction representing the potentially occur in an experiment by the
number of times the event did occur. In this case,
number of outcomes that could possibly happen
16-10 is 6, and 6 will be our numerator. We will
in the experiment based on what the
use the same denominator as we did for the
experiment would look like if you repeated in
probability for the number of times D’Andrus
flipped “heads” because the number of trials does
infinity times. It is different from experimental
not change. We now have the answer 6/16, which probability because it is based off of how many
can be simplified to 3/8. The experimental times an event COULD occur in the experiment,
probability that D’Andrus flipped a “tail” is 3/8. not the number of times the event DID occur in
the experiment. In other words, you are
Problem 2: You spin a spinner with sectors labeled biasing theoretical probability based on odds,
green, yellow, blue, orange, and red. You spin red not what happened in an experiment.
once, yellow twice, blue three times, orange six
times, and red once. What was your experimental Problem 1: There are 9 blue sheets of paper,
probability of spinning orange or blue? 10 green sheets of paper, 2 white sheets of
paper, 4 yellow sheets of paper, and 5 black
9/13
sheets of paper in a box and I pulled out one
sheet of paper randomly, what is the theoretical
probability that I would select a white or green
sheet of paper?

5. Tree Diagrams and the
Counting Principal
To find out the number of possible outcomes A tree diagram is a diagram that helps
that could occur, you must add up all of the you see all of the possible outcomes in a set. It
events that occurred and the number of times consists of branches and the categories increase
each event could occur. When you add up all and get more specific as you go down. An
of the events and numbers of times each event example of a tree diagram is included below.
could occur (9+10+2+4+5), you get a sum of
30. Now you know that your probability’s
fraction’s denominator is 30 because the
number of trials in this set is 30. The favorable
events we want to occur for our numerator are
green and white. White occurred twice, and
green occurred 10 times. These two values
must be put together to reach 12 on the
numerator because 12 out of the 30 trials are
favorable events. We now have the fraction
12/30, or 6/15, which is our probability.

6. The counting principal refers to how to find a*b because there are more events. Since
the occurrence of more than one activity. each of the 10 events have 4 outcomes, we
The rule for the fundamental counting multiply 4*4*4*4*4*4*4*4*4*4, or 4 to
principal is that if there are a possible ways the 10th power. The product is 1,048,576.
to choose the first item and b possible ways This means that the odds of getting a
to choose the second item (after the first perfect score on the test would be
item was chosen), then there are a*b ways 1/1,048,576. The numerator is 1 because
to choose all of the items combined. This there is only one combination of answers
also works for more than one event. to choose that will get you a perfect score,
or only one favorable outcome.
Problem 1: If I am given a 10 question
multiple choice (A, B, C, D) quiz, and I Problem 2: An ice cream shop offers 31
guess on every single question, what is the flavors and 5 toppings. How many
probability of scoring a perfect 100%? different types of ice cream could be made
with one flavor and one topping?
In this case, there are 10 events as
opposed to 2, but the concept of the 155 different types of ice cream
fundamental counting principal still
applies. However, our expression will look
like: a*b*c*d*e*f*g*h*i*j instead of just

7. Independent and Dependent Events:
Dependent Events There are 52 cards in a deck. What is the probability
of getting two “9”s in a row?
If the events are independent, the existence of
If this was an independent event, I would multiply 4/52
one event does not affect the probability of the other
by 4/52, but this is a dependent event, meaning the
event. If the events are dependent, the existence of one
outcome of the 1st event will affect the outcome of the
event does affect the probability of the other event. An
second event. Since one card is being pulled out of the
example of an independent event is rolling a “3” on a
deck, the second time I’m pulling out a card, there are
fair number cube on one roll and a “4” on the next roll.
only 51 cards or 51 trials, meaning my denominator
An example of a dependent event is the amount of time
changes to 51. There is also one less “9” cards in the
a student spends studying for a test and the probability
deck because I removed one of the “9” cards from the
of them receiving a 90 or above on the test.
deck when I picked my first card. Since there is one
less event that can occur, my numerator changes to 3.
Independent Events
So, instead of doing 4/52*4/52, we will multiply
4/52*3/51. The product is 12/2652, which can be
Problem 1: What is the probability of rolling a 6 on a
simplified to 1/221.
fair number cube twice in a row?
Problem 2: A gumball machine has 20 red, 20 green,
Since this is an independent event, you can multiply
20 blue, and 20 pink gumballs. Dennis then takes a
1/6 by 1/6 (fundamental counting principal) to
pink gumball from the machine. Then, Mark takes a
determine the answer. The answer is 1/12.
gumball. What is the probability that the gumball he
chooses will be red?
Problem 2: What is the probability of tossing “heads”
on a coin twice in a row? 20/79
1/4

8. Mark Winokur
9/23/11
Green
Probability Guide Book
Probability
guide
book
By Mark
Winokur