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Intro to Probability, Sample spaces, and probabilty distributions

posted Mar 4, 2010, 10:30 AM by Prof Kiernan   [ updated Mar 23, 2010, 4:33 PM ]
·         A sample space is a “set of all possible outcomes or events in an experiment that cannot be further broken down” (Triola, 2008).

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A probability distribution is “a collection of values of a random variable along with their corresponding probabilities” (Triola, 2008).


Common confusion about probability distributions:

  • There are no negative probabilities.
  • There are no probabilities larger than one.
  • The total of all probabilities must add up to one.

Common confusion about sample spaces:

  • A sample space has no probabilities.

 

Factor tree method

 

Example 1: If we wanted to find the probability of flipping a coin 3 times and having 0,1,2 or 3 heads we could use a factor tree to show the possible combinations.

To create a factor tree, we must start with the first possibility (heads or tails in blue). Then the second coin possibilities (in red) branch off of each of the first coin toss. Finally, the third coin (in green) branches off of each of the second coin toss.


 

 

Sample space:

S = { Heads Heads Heads, Heads Heads Tails, Heads Tails Heads, Heads Tails Tails,

Tails Heads Heads, Tails Heads Tails, Tails Tails Heads, Tails Tails Tails}


Probability distribution:


Combinations

Number of heads (x)

P (x )

Tails Tails Tails

0

1/8

Heads Tails Tails,

Tails Heads Tails,

Tails Tails Heads

1

3/8

Heads Heads Tails,

Heads Tails Heads,

Tails Heads Heads

2

3/8

Heads Heads Heads,

3

1/8

 


Example 2:

If you were to have 2 children, what would the probability of having zero, one, or two boys be?

To answer this question the first thing we need is to define the sample space. That is, we need to create a list of all possible combinations of children. Illustrate that you can have a boy then another boy, or a boy then a girl, or a girl then a boy, or a girl then another girl.

 

S = {Boy Boy, Boy Girl, Girl Boy, Girl Girl}

Next write the probability distribution on the board showing where each of the probabilities come from (combinations column):

Combinations

Number of boys in 2 births (x)

P (x)

Girl Girl

0

¼

Boy Girl

Girl Boy

1

½

Boy Boy

2

¼

 

 

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