Cumulative Probabilities Worksheet
There are some examples you may wish to examine before
attempting these exercises.
 Note
Interpret <= as less
than or equal to (at
most, or no more than).
Exercise 1
Here's the probability distribution for a
discrete random variable X
x 
f(x) 
P[
X <=
x ] 
0 
0.15 

1 
0.35 

2 
0.25 

3 
0.15 

4 
0.10 

 Find the mean and standard deviation for this
variable.
 Compute and table the cumulative probabilities in
the third column of the table.
Exercise 2
Here's the probability distribution for X
= number of children in a randomly selected family. The
mean is 1.6094. Compute the and table the cumulative
probabilities in the third column.
x

f(x) 
P[ X
<= x
] 
0 
0.20000 

1 
0.32189 

2 
0.25903 

3 
0.13896 

4 
0.05591 

5 
0.01800 

6 
0.00483 

7 
0.00111 

8 
0.00022 

9 
0.00004 

10 
0.00001 

Exercise 3
Based on recent records, the manager of a
car painting center has determined the following
cumulative probabilities for X = the number of
customers per day.
x 
P[ X <= x ] 
0 
0.05 
1 
0.25 
2 
0.55 
3 
0.80 
4 
0.95 
5 
1.00 
 What is the probability the center serves at most
3 customers in a day?
 What is the probability the center serves less
than 3 customers in a day?
 What is the probability the center serves exactly
3 customers in a day?
 What is the probability the center serves at most
3 customers in a day?
 What is the probability the center serves more
than 3 customers in a day?
 What is the probability the center serves no
customers in a day?
 Find the probability X is between 2 and 4
inclusive. That is, find P[2<=X<=4].
 Find P[ 1<=X<=3 ].
Exercise 4
When a fair coin is tossed 6 times, and X
= number of heads is counted, then the cumulative
probabilities are as tabled below.
x 
P[ X <= x ] 
f(x) 
0 
0.0156 

1 
0.1094 

2 
0.3438 

3 
0.6563 

4 
0.8906 

5 
0.9844 

6 
1.0000 

 Find the probability there are at most 4 heads.
 Find the probability there are exactly 4 heads.
 Find the probability there are less than 4 heads.
 Find the probability there are more than 4 heads.
 Find the probability there are at least 4 heads.
 Find the probability there are between 1 and 5
heads, inclusive.
 What's the probability each of the 6 tosses comes
up the same?
 Look on page 657 of your textbook. Use the row
labeled n = 6 (there are n = 6
trials, or tosses of the coin) and the column
labeled p = 0.50 (the probability of a
head in any one trial is p = 0.50).
Compare to the table above.
 Fill in the column of probabilities f(x).
Sum these.
 Find the mean of X. Verify that this makes
sense. Rumor has it that there's a shortcut for
computing the mean in this circumstancemerely
find np. Is this true?
 Find the variance and standard deviation of X.
Rumor has it that there's a shortcut for
computing the variance in this
circumstancemerely find np(1  p). Is
this true?
