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Probability
describes the chance that an
uncertain event will occur. |
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Empirical Probability of an
event is an "estimate" that the event will happen based on how often the
event occurs after collecting data or running an experiment (in a large
number of trials). It is based specifically on direct observations
or experiences.
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Empirical
Probability Formula
P(E) = probability that an event, E,
will occur.
top = number of ways the specific event occurs.
bottom = number of ways the experiment
could occur. |
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Example: A
survey was conducted to determine students' favorite breeds of
dogs. Each student chose only one breed.
| Dog |
Collie |
Spaniel |
Lab |
Boxer |
PitBull |
Other |
| # |
10 |
15 |
35 |
8 |
5 |
12 |
What is the probability that a student's favorite dog breed
is Lab?
Answer: 35 out of the 85
students chose Lab. The probability is
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Theoretical Probability
of an event is the number of ways that
the event can occur, divided by the total number of outcomes.
It is finding the probability of events
that come from a sample space of known equally likely outcomes.
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Theoretical
Probability Formula
P(E) = probability that an event, E,
will occur.
n(E) = number of equally likely outcomes of E.
n(S) = number of equally likely outcomes of
sample space S. |
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Example 1:
Find the probability of rolling a six on a fair die.
Answer: The sample space for
rolling is die is 6 equally likely results: {1, 2, 3, 4, 5, 6}.
The probability of rolling a 6 is one out of 6 or
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Example 2:
Find the probability of tossing a fair die and getting an odd
number.
Answer:
event E : tossing an odd number
outcomes in E: {1, 3, 5}
sample space S: {1,
2, 3, 4, 5, 6} |
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Comparing Empirical and Theoretical
Probabilities:
Karen and Jason roll
two dice 50 times and record their results in the
accompanying chart.
1.) What is their empirical probability of
rolling a 7?
2.) What is the theoretical probability of
rolling a 7?
3.) How do the empirical and theoretical
probabilities compare? |
| Sum of the
rolls of two dice |
3, 5, 5,
4, 6, 7, 7, 5, 9,
10,
12, 9, 6, 5, 7, 8,
7, 4, 11, 6,
8, 8, 10, 6, 7, 4,
4, 5, 7, 9,
9, 7, 8, 11, 6, 5,
4, 7, 7, 4,
3, 6, 7, 7, 7, 8,
6, 7, 8, 9
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Solution:
1.) Empirical probability (experimental probability or
observed probability) is 13/50 = 26%.
2.) Theoretical probability (based upon what is
possible when working with two dice) = 6/36 = 1/6 = 16.7%
(check out the table at the right of possible sums when
rolling two dice).
3.) Karen and Jason rolled more 7's than would be
expected theoretically. |
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Vocabulary to keep in mind:
Biased:
one result has a better chance of happening than
another result.
Unbiased (fair) :
each result has an equal chance of happening.
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