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Probability Experiments
When weather forecaster say that there is 90 % chance of rain or a physician says that there is a 35 % chance for a successful surgery, they are stating the chance or probability that a specific event will occur. Decision such as “should you wash your car”, or should you opt for surgery” are often based on these probabilities.
Definition:
A probability experiment is an action, or trial, through which specific results (counts, measurements, or responses) are obtained, the result of a single trial in a probability experiment is an outcome. The set of all possible outcomes of a probability experiment is the sample space. An event consists of one or more outcomes and is a subset of the sample space.
Types of probability
There are three types of probability: Classical Probability, Empirical Probability and Subjective Probability. The probability that even ‘E’ will occur is written as P (E) and is read as “the probability of even E”
Definition
Classical (or theoretical) probability is used when each outcome in a sample space is equally likely to occur. The classical probability of en event E is given below by a formula:
E = Number of outcomes in event E / Total Number of outcomes in a sample space
Example:
Rolling a six sided die find the probability of
1. Event A; Rolling a 3
2. Event B : Rolling a 7
3. Event C : Rolling a number less than 5
Solution:
When you roll a six sided die the sample space consists of six outcomes
[1, 2, 3, 4, 5, 6]
1. There is one outcome in event A {3}
P (3) = I/6 = 0.167
2. As 7 is not in the sample space so there is no outcome in event B, So :
P (7) = 0/6 = 0
3. There are four outcomes in event C: [1, 2, 3, and 4] so:
P (Number less than 5) = 4/6 = 0.667
A second type of probably is empirical (or Statistical)probability, empirical probability can be used even if each outcome is not equally likely to occur.
Definition
Empirical probability:
Empirical (or statistical probability) is based on observations obtained from probability experiments. The empirical probability of an event E is relative frequency of event E
P (E) = Frequency of event E / Total Frequency
= f / n
Example:
A pond contains three types of fish: Blue-gills, Red-gills and Crappies. Each fish in this pond is equally likely to be caught. You caught 40 fish and record the type. Each time, you release the fish back into the pond. The following frequency distribution shows your results.
Bluegill……………= 13
Red-gill……………= 17
Crappy……………..= 10
Total Frequencies. …= 40
If you catch another fish, what is the probability that it is a bluegill?
Solution The event is “catching a bluegill”. In your experiment, the frequency of this event is 13. Because the total of the frequencies are 40, the empirical probability of catching a bluegill is
P (Bluegill) 13/40 =0.325
Subjective Probability
The subjective probabilities result from intuitions, educated guesses, and estimates. For example given a patient’s health and extent of injuries, the doctor may feel that the patient has 90 % chance of full recovery. Or a business analyst may predict that the chance of the employees of a certain company going on strike is 0.25
Example
Classifying types of probability
Classify each statement as an example of classical probability, empirical probability, or subjective probability explain your reasoning
1. The probability of ringing your phone during dinner is 0.5
2. The probability that a voter chosen at random will vote Pakistan People party is 0.45
3. The probability of winning a 1000- ticket raffle with one ticket is 1/1000
Solution
1. This probability is most likely by educated guess, it is an example of subjective probability
2. This statement is most likely based on survey of a sample of voters , so it is an example of empirical probability
3. Because you know the outcomes and each outcome is equally likely to occur, this is an example of classical probability
To be continued……………………..
When weather forecaster say that there is 90 % chance of rain or a physician says that there is a 35 % chance for a successful surgery, they are stating the chance or probability that a specific event will occur. Decision such as “should you wash your car”, or should you opt for surgery” are often based on these probabilities.
Definition:
A probability experiment is an action, or trial, through which specific results (counts, measurements, or responses) are obtained, the result of a single trial in a probability experiment is an outcome. The set of all possible outcomes of a probability experiment is the sample space. An event consists of one or more outcomes and is a subset of the sample space.
Types of probability
There are three types of probability: Classical Probability, Empirical Probability and Subjective Probability. The probability that even ‘E’ will occur is written as P (E) and is read as “the probability of even E”
Definition
Classical (or theoretical) probability is used when each outcome in a sample space is equally likely to occur. The classical probability of en event E is given below by a formula:
E = Number of outcomes in event E / Total Number of outcomes in a sample space
Example:
Rolling a six sided die find the probability of
1. Event A; Rolling a 3
2. Event B : Rolling a 7
3. Event C : Rolling a number less than 5
Solution:
When you roll a six sided die the sample space consists of six outcomes
[1, 2, 3, 4, 5, 6]
1. There is one outcome in event A {3}
P (3) = I/6 = 0.167
2. As 7 is not in the sample space so there is no outcome in event B, So :
P (7) = 0/6 = 0
3. There are four outcomes in event C: [1, 2, 3, and 4] so:
P (Number less than 5) = 4/6 = 0.667
A second type of probably is empirical (or Statistical)probability, empirical probability can be used even if each outcome is not equally likely to occur.
Definition
Empirical probability:
Empirical (or statistical probability) is based on observations obtained from probability experiments. The empirical probability of an event E is relative frequency of event E
P (E) = Frequency of event E / Total Frequency
= f / n
Example:
A pond contains three types of fish: Blue-gills, Red-gills and Crappies. Each fish in this pond is equally likely to be caught. You caught 40 fish and record the type. Each time, you release the fish back into the pond. The following frequency distribution shows your results.
Bluegill……………= 13
Red-gill……………= 17
Crappy……………..= 10
Total Frequencies. …= 40
If you catch another fish, what is the probability that it is a bluegill?
Solution The event is “catching a bluegill”. In your experiment, the frequency of this event is 13. Because the total of the frequencies are 40, the empirical probability of catching a bluegill is
P (Bluegill) 13/40 =0.325
Subjective Probability
The subjective probabilities result from intuitions, educated guesses, and estimates. For example given a patient’s health and extent of injuries, the doctor may feel that the patient has 90 % chance of full recovery. Or a business analyst may predict that the chance of the employees of a certain company going on strike is 0.25
Example
Classifying types of probability
Classify each statement as an example of classical probability, empirical probability, or subjective probability explain your reasoning
1. The probability of ringing your phone during dinner is 0.5
2. The probability that a voter chosen at random will vote Pakistan People party is 0.45
3. The probability of winning a 1000- ticket raffle with one ticket is 1/1000
Solution
1. This probability is most likely by educated guess, it is an example of subjective probability
2. This statement is most likely based on survey of a sample of voters , so it is an example of empirical probability
3. Because you know the outcomes and each outcome is equally likely to occur, this is an example of classical probability
To be continued……………………..
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