Probability Distribution:
A probability distribution is an assignment of probabilities to the specific values of the
random variable or to a range of values of the random variable.
A random variable has a probability distribution whether it is discrete or continuous.
The probability distribution of a discrete random variable has a probability assigned to
each value of the random variable. The sum of these probabilities must be 1.
In a probability distribution, your probabilities column is calculated the same way we did
relative frequency.
???
Frequency
Σ=
Probability
Σ=
Let’s look at an example:
A recent poll was taken of 125 participants. The poll asked on a scale of 1 – 6,
rate McFadden’s Restaurant, with 1 being the worst and 6 being the best. The table
below gives the results of the poll:
Rating
1
2
3
4
5
6
Frequency
12
11
28
18
19
37
Σ=
Probability
Σ=
To fill in the rest of the table, we just take each frequency for the given rating and
divide it by the total number of participants.
We can then graph the probability distribution as a histogram with probability on
the vertical axis and rating on the horizontal axis.
Examples:
1) Complete the table below and graph the results:
1
2
Frequency
65
78
Probability
3
4
111
176
Σ=
Σ=
2) Complete the table below and graph the results:
Rating
0
1
2
3
4
Frequency
245
267
222
198
210
Probability
Σ=
Σ=
3) A psychologist has devised a test that measures the congeniality factor (desire to
work with people). A random sample of 1,000 adults took the test. Their scores
are given in the following table:
x (Score)
1
2
3
4
5
Frequency of Adults with this Score
90
480
220
150
60
a) Construct a table to show the probability distribution.
b) Construct a histogram to display the data.
4) The head nurse on the third floor of a community hospital is interested in the
number of nighttime room calls requiring a nurse. For a random sample of 208
nights (9:00 P.M. – 6:00 A.M.) the following information was obtained where x =
the number of room calls requiring a nurse and f = frequency with which this
many calls occurred (i.e., number of nights).
x
f
36
6
37
10
38
11
39
20
40
26
41
32
42
34
43
28
44
25
45
16
a) If a night is chosen at random from these 208 nights, use relative
frequencies to find P(x) when x = 36, 37, 38, 39, 40, 41, 42, 43, 44,
and 45.
b) Use a histogram to graph the probability distribution of part a.
c) Assuming these 208 nights represent the population of all nights at
community hospital, what do you estimate the probability is that on a
randomly selected night there will be from 39 to 43 (including 39 and
43 room calls requiring a nurse?
d) What do you estimate the probability is that there will be from 36 – 40
(including 36 and 40) room calls requiring a nurse?
5) The army gives a battery of tests to all new recruits. One exam measures a
person’s ability to work with technical machinery. This exam was given to a
random sample of 360 new recruits. The results are shown below, where x =
score on a ten point scale and f = frequency of new recruits with this score.
x
f
1
28
2
42
3
79
4
83
5
51
6
36
7
18
8
12
9
7
10
4
a) If a person is chosen at random from those taking the test, use the
relative frequency of test scores to estimate P(x) as x goes from 1 to
10.
b) Use a histogram to graph the probability distribution of part a.
c) The ground-to-air missile battalion needs people with a score of seven
or higher on the exam. What is the probability that a new recruit will
meet the requirement?
d) The kitchen battalion can use people with a three or less. What is the
probability that a new recruit is not overqualified for this work?