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Thinking Ahead about Probability 1 – You should understand these questions fully before the next class.
Check your answers with the key on your instructor’s website. You can get help with this work from the
following sources:
• Visit your instructor during office hours
• go to the MAC (Math Assistance Center) 700 BH
• go to http://www.mathgoodies.com/lessons/vol6/intro_probability.html
Example 1:
Solution:
You throw a single six-sided die. What is the probability of rolling a 4?
The probability of rolling a 4, abbreviated as P(4), is 1 in 6, or one sixth.
Definitions:
• An Outcome is a possible result of a measurement or an experiment.
• The Desired Outcome of the experiment above was rolling a 4.
• The Sample Space is the set of all possible distinct outcomes.
• An Event is an outcome or a set of outcomes of a measurement or experiment.
• The Desired Event above was rolling a 4.
• The Multiplicity of an event is the number of possible ways the event may occur.
• Mutually Exclusive Events are events that cannot happen at the same time.
• The Probability of the occurrence of an event is the measure of the chance that the event will happen. If all
possible outcomes are equally likely, the probability, indicated by P, is:
P=
the number of ways the event may occur
the total number of possible outcomes
Since the total number of possible outcomes (denominator) will always be greater than or equal to the number
of ways an event may occur (numerator) the probability is always a number between zero and one.
0 < P (event ) ≤ 1 .
For the case of the six-sided die, the sample space is 1,2,3,4,5,6, so there are six distinct possible outcomes,
only one of which is a 4, so P(4) = 16 = 16 23 % .
The Multiplicity of an event is the number of possible ways the event may occur. In the next problem, the
multiplicity of the desired event is 3.
Example 2:
Solution:
A single 6-sided die is rolled. What is the probability of rolling an even number?
The desired event is any one of 3 even-numbered outcomes (2, 4, or 6) out of a total number
of six possible outcomes, so the probability is 3 out of 6 or ½ = 50% for an even-numbered
outcome.
Note that in the second problem, the probability of the event (even-numbered outcome) was equal to the sum
of the probabilities of the individual even-numbered outcomes:
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P(even number) = P(2) + P(4) + P(6) =
1 1 1 3 1
+ + = =
6 6 6 6 2
Note that the sum of the probabilities for all possible outcomes is always 1 or 100%, since it is the total
number of possible outcomes divided by the total number of possible outcomes.
A corollary to this is that the probability that a particular event will not happen is one minus the probability
that the even will happen:
P ( event will not happen ) = 1 − P ( event ) .
Exercises
1. A six-sided die is rolled. What is the probability of getting a number less than 3?
2. A six-sided die is rolled. What is the probability of not rolling a 5?
3. What is the probability of choosing a vowel by choosing one letter randomly from our 26-letter alphabet?
(Do not count y as a vowel.)
Example 3:
A sock drawer contains 5 red pairs, 5 black pairs, 3 white pairs, and 7 blue pairs of socks. If a
pair is chosen randomly from the drawer, what is the probability of choosing a black pair?
Outcomes:
The sample space of this experiment is red, black, white and blue. The total number of
possible outcomes is 20.
The multiplicities of each possible event are: The probabilities are:
Choosing Red
=
5
P(red)
=
5/20 = 25%
Choosing Black =
5
Choosing White
=
Choosing Blue =
7
Solution:
P(black)=
3
5/20 = 25%
P(white)
P(blue)
=
=
3/20 = 15%
7/20 = 35%
The probability of the desired event (choosing a black pair) is 25%.
Note that the probabilities add up to 100%. The most probable random choice is a blue pair of socks, and the
least probable is a white pair. Randomly choosing a red or a black pair is equally likely since P(black) = P(red).
Problem 4:
Solution:
If one pair of socks is chosen randomly from the sock drawer above, what is the probability of
choosing either a black pair or a white pair?
These are two mutually exclusive events since, in choosing only one pair, if we choose a black
pair, we cannot also choose a white pair. When two events are mutually exclusive, the
probability that event 1 or event 2 will occur is the sum of the probability of each event.
P ( event 1 or=
event 2 ) P ( event 1) + P ( event 2 )
In this case the probability of choosing either a black pair or a white pair is 40%.
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Exercises
4. For choosing one pair randomly from the sock drawer above, what is the probability of not choosing a blue
pair or a red pair?
5. What is the probability of choosing the letter “s” by randomly choosing a letter from the word Mississippi?
What is the sample space?
6. A card is selected randomly from an ordinary deck of 52 cards. What is the probability of selecting an 8 or a
black king or a red 10?
7. How many possible outcomes are there for two dice rolled at the same time?
8. What is the probability of obtaining the sum of 5 for two dice rolled at the same time?