Theoretical vs. Empirical Probability

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Theoretical vs. Empirical Probability

Probability-
describes the chance that an uncertain event will occur.

Empirical Probability - estimate that the event will happen based on how often the event occurs after collecting the data or running an experiment. It is based specifically on direct observation or experiences.

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.

Example: A survey was conducted to determine students' favorite color. Each student chose only one color. colorRedBlueBlackYellowPurpleOther
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What is the probability that a student's favorite color being black? Answer: 35 out of the 85 students chose Black. The probability is
.

Theoretical Probability-Theoretical Probability: is the number of ways that an event can occur. You need to divide by the total number of outcomes. This is usually used with equally likely out comes.

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. 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

.
Example 2: Find the probability of tossing a 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}

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

Comparing Empirical and Theoretical Probabilities:
Karen and Jason roll two dice 50 times and record their results in the...
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