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Making Predictions Based on Theoretical Probability
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The probability of an event (E) occurring is the ratio of the number of ways that
event can occur over the total number of outcomes in the sample space. The
sample space is every possible outcome. The probability of an event occurring is
always between 0 and 1. The probability of 0 means that the event is
impossible. A probability of 1 means that the event is certain to happen.
P(event ) =
number of ways event can occur
number of outcomes in sample space
0 ≤ P(E) ≤ 1
Example of Probability
Find the probability of choosing a red marble out of a bag if there are 4 red
marbles, 3 yellow marbles, and 3 green marbles.
Event = choosing a red marble
Number of ways to choose a red marble = 4
Sample space = total number of marbles
Number of outcomes in sample space = 10
P(red) =
number of ways an event can occur
number of outcomes in sample space
P(red) =
4
= 0.4
10
If there is an experiment in which we want to find out how likely it is that two
events happen at the same time, like choosing multiple marbles from a bag of
marbles, then we need to determine if the second outcome is affected by whether
or not the first outcome is replaced. Either the first outcome is replaced or it is
not replaced. Below are examples of events that occur with replacement and
without replacement.
With Replacement
With replacement means that the object chosen in the first event is returned to
the sample space before the 2nd event occurs. Since the object is returned, the
probability of the second event is not affected.
Example
From the same bag of marbles (4 R, 3 Y, 3 G), one red marble is chosen and
then returned into the bag. Then another marble is chosen. What is the
probability of the 2nd marble chosen being red?
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Answer: Since the first red marble chosen is retuned to the bag, there
are still 10 marbles in the bag when the second marble is chosen. Plus,
there are still 4 red marbles. So, the probability of the 2nd marble chosen
being a red marble is 4 out of 10 (which can reduce to 2 out of 5, or
2/5).
Without Replacement
Without replacement means that the object chosen in the first event is not
returned to the sample space before the 2nd event occurs. This, in turn, affects
the probability of the second event.
Example
From the same bag of marbles (4 R, 3 Y, 3 G), one red marble is chosen and not
returned into the bag. Then another marble is chosen. What is the probability of
the 2nd marble chosen being red?
Answer: Since the first red marble chosen is not retuned to the bag,
there are only 9 marbles in the bag when the second marble is chosen.
Plus, there are now only 3 red marbles to choose. So, the probability of
the 2nd marble chosen being a red marble is 3 out of 9 (which can reduce
to 1 out of 3, or 1/3).
To predict the outcome of a probability experiment, you need to be able to
determine the sample space and the number of possible outcomes that satisfy
the event. The probability is just the number of outcomes that satisfy your event
over the number of total outcomes in the sample space. This is also referred to
as theoretical probability.
P(event ) =
number of ways an event can occur
number of outcomes in sample space
Below are two examples of how to predict outcomes.
Example
A card is pulled from a standard deck of cards. What is the probability that it is
the 8 of hearts?
8
Answer: First off, we need to determine the sample space.
The sample space in this problem is each card in a deck.
There are 52 cards in a deck. So, we know the number of
outcomes in the sample space is 52. Now, we have to
determine how many 8 of hearts are in a deck. There is only
one 8 of hearts. So the probability of choosing the 8 of
hearts is 1 out of 52.
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8
Example
After this die is rolled, what is the probability that the number on the top is
greater than 4?
Answer: First off, we need to determine the sample space.
The sample space in this problem is each face on the cube
(numbered 1, 2, 3, 4, 5, and 6). There are 6 faces on a cube.
So, we know the number of outcomes in the sample space
is 6. Now, we have to determine how many numbers greater
than 4 could be on the top. Only 5 and 6 are greater than 4.
So, we know the number of ways we can roll a number greater than 4 is
2. Therefore, the probability of rolling a number greater than 4 is 2 out
of 6, which reduces to 1 out of 3.
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