2.0 Outline
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Definitions (random exp., sample space, event,...)
Probability (definition, requirements,...)
Discrete probability distribution
Continuous probability distribution
2. Probability and Probability Dis.
2.1 Random Experiments
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An …….., ……., or ………….. that leads to one of the several ………, even though it is repeated in the same manner.
Random experiment Outcome
Flip a coin
Record marks on a course test (out of 100) Record student evaluations of a course Toss a die
2.2 Sample Space
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A list of all ...of a ...is called the sample space of the experiment.
𝑆 = 𝑂1, 𝑂2, ⋯ , 𝑂𝑛 (2.1)
It is defined based on the objective of the analysis.
The outcomes must be exhaustive and mutually exclusive.
Random experiment Sample space
Flip a coin
A part conforms to the specifications Record student evaluations of a course Toss a die
The positions of two switches in series
2. Probability and Probability Dis.
2.3 Event
A subset of the sample space of a random experiment is called an event.
Event The event set
Number is even in tossing a fair die The summation is 5 in rolling 2 dice The circuit is closed in two switches in series The circuit is open in two switches in series
2.4 Random Variable
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The word random reminds that we do not usually know what its value is until we observe or perform the experiment.
A random variable is denoted by an uppercase latter (e.g, X), while its measured value is denoted a lowercase letter (e.g., x=2).
……….
……….
……….
2. Probability and Probability Dis.
2.4.1 Discrete Random Variable
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A discrete random variable has either a ………
of values, or a ………of values, that is, they result from a counting process.
• The number of students absent from statistics class today.
• The number of defective parts among 2000 tested.
……….
……….
……….
2.4.2 Continuous Random Variable
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A continuous variable has ……….. of values (………..) that can be associated with measurements on a continuous scale in such a way that there are no gaps or interruptions.
• Density and viscosity of a chemical compound
……….
……….
……….
2. Probability and Probability Dis.
2.5 Probability concept
A critical component of inference is probability, because it gives the link between the sample and the population.
The probability of an event can be determined by three approaches:
……….
……….
……….
2.5.1 Probability: Classical approach
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This approach is used by mathematicians.
The approach requires equally likely outcomes.
• n :
• n(A) :
……….
2. Probability and Probability Dis.
2.5.2 Probability: Relative frequency approach
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A random experiment is conducted (observed) a large number of times.
Number of times that an event Aoccurs is counted.
Method only gives an approximation of an exact value
Law of large numbers: ………..
……….
……….
2.5.3 Probability: Subjective approach
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The classical approach is not possible to use.
No history of the outcomes.
The degree of belief that we hold in the occurrence of an event, is used to approximate probability.
2. Probability and Probability Dis.
2.6 Probability distribution
A probability distribution is a ……., ……., or ………
that describes the possible values of a random variable along with the probabilities associated with these values.
There are two types of distribution:
• Discrete probability distribution
• Continuous probability distribution
2.6.1 Discrete probability distribution
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For a discrete random variable, the associated probability is a discrete probability distribution.
Every discrete probability distribution must satisfy two fundamental requirements for each value:
Can formula 𝑃(𝑥) = 𝑥/5 for 𝑥 =1,2 and 3 be determining a probability distribution? Why?
……….
……….
2. Probability and Probability Dis.
2.6.1.1 Dis. P. D.: Binomial
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The outcomes belong to either of two relevant categories.
A binomial probability distribution results from a binomial experiment with the following requirements:
• On each trial there are 2 possible outcomes
……….
……….
……….
……….
2.6.1.1.1 Dis. P. D.:Binomial-Exp.
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•Flip a coin 15 times. The two outcomes in each trial are heads and trials. 𝑃(𝑋: 𝑠𝑢𝑐𝑐𝑒𝑠𝑠) = 𝑝=0.5
•A drug can either cure or not cure a patient. A certain drug has probability 0.9 of curing a disease. It is administered to 100 patients. (n=100, p=0.9).
•A manufacturer of computer chips finds that on the average 5% are defective. They take a random sample of size 75. If the sample contains more than 6 defective chips, then the process is stopped. (n=75, p=0.05).
2. Probability and Probability Dis.
2.6.1.1.2 Dis. P. D.: Binomial-Formula
Binomial distribution can be determined by:
n: number of trials
x: number of successes among n trials
p: probability of success in one trial, which is constant
Cumulative binomial probability
……….
2.6.1.1.3 Dis. P. D.: Binomial-Table
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Also binomial distribution can be determined by the associated table:
2. Probability and Probability Dis.
2.6.1.1.4 Dis. P. D.: Binomial-Exp
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n=20 , p=0.5 ● n=20 , p=0.2 ●
