Section 4.9: Solving Probability Problems by
Using Combinations
This section combines the counting skills we
learned in the last two sections with the
probability skills we learned earlier in the
chapter. This section can be really tricky if you
don’t completely understand everything we
have done in the chapter so far. If you don’t
feel comfortable with something we did earlier
in the chapter visit me to get things straight
before proceeding.
Here is a probability formula that we used
earlier in this chapter. The formula isn’t new,
but in this section the numbers that we put in
the numerator as well as the number we put in
the denominator will contain combinations.
Here is a basic probability formula that we have
already used:
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑐𝑜𝑚𝑒
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡ℎ𝑒 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟
=
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑤𝑖𝑡ℎ 𝑛𝑜 𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛
Example: A club consists of four men and five
women. Three members are to be selected at
random to form a committee. What is the
probability that the committee will consist of
three women? (Write your answer as a percent
and round to 1 decimal.)
The order in which the three members are
selected is not important (since we are not
attaching labels like President, Vice-President
and Secretary) so we may use combinations.
P(committee consists of three women) =
Total number of 3 member committees
I need 3 of 5 women for the numerator so the
numerator will be 3C5.
I need any 3 of the 9 club members for the
denominator. The denominator will be 9C3.
5C3
9C3
=
10
84
= 11.9%
So the probability of the three person
committee that is randomly selected from a
club of four men and five women being
composed of only women is about 11.9 %.
Answer: 11.9%
Example: 2 cards are drawn from a standard
deck of cards. What is the probability they are
both red? (Write your answer as a percent and
round to 1 decimal.)
For numerator I need 2 of the 26 red cards in a
deck. The numerator will be 26C2.
For the denominator I can get any 2 of the 52
cards in a deck. The denominator will be 52C2.
𝑃(𝑏𝑜𝑡ℎ 𝑐𝑎𝑟𝑑𝑠 𝑎𝑟𝑒 𝑟𝑒𝑑) =
=
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑜𝑢𝑝𝑠 𝑜𝑓 2 𝑜𝑓 26 𝑟𝑒𝑑 𝑐𝑎𝑟𝑑𝑠
26C2
=
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑜𝑢𝑝𝑠 𝑜𝑓 2 𝑜𝑓 52 𝑡𝑜𝑡𝑎𝑙 𝑐𝑎𝑟𝑑𝑠 52C2
325
1326
Answer: about 24.5%
Homework #1-16 (Write your answer as a
percent rounded to 1 decimal place.)
1) A class consists of 19 girls and 15 boys. If 5
of the students are to be selected at random,
determine the probability they are all girls.
2) A club has 8 boys and 5 girls. 3 members are
selected at random, determine the probability
they are all boys.
3) Of 35 people attending a dance 28 have a
college degree. If 4 people at the dance are
selected at random, find the probability each
has a college degree.
4) Of 50 people attending a concert 30 have
seen the group before. If 8 people at the
concert are selected at random, find the
probability each has seen the group before.
5) A class of 16 people contains 4 people
whose birthday is in October. If 3 people from
the class are selected at random, find the
probability that none of those selected has an
October birthday.
6) A class of 25 people contains 5 Gemini’s. If 4
people from the class are selected at random,
find the probability that none of those selected
are Gemini’s.
7) A committee of 4 is to be randomly selected
from a group of 7 teachers and 8 students.
Find the probability the committee will consist
of 4 students.
8) A committee of 4 is to be randomly selected
from a group of 7 teachers and 8 students.
Find the probability the committee will consist
of 4 teachers
9) Anthony’s wallet contains 8 bills of the
following denominations, four $5 bills, two $10
bills, one $20 bill and one $50 bill. If he selects
two bills at random, determine the probability
that he selects 2 $5 bills.
10) Anthony’s wallet contains 8 bills of the
following denominations, four $5 bills, two $10
bills, one $20 bill and one $50 bill. If he selects
two bills at random, determine the probability
that he selects 2 tens.
11) 3 cards are drawn from a standard deck of
cards. What is the probability they are all
hearts?
12) 4 cards are drawn from a standard deck of
cards. What is the probability they are all
clubs?
13) 2 cards are drawn from a standard deck of
cards. What is the probability they are both
red?
14) 2 cards are drawn from a standard deck of
cards. What is the probability they are both
black?
15) 2 cards are drawn from a standard deck of
cards. What is the probability they are all
queens?
16) 2 cards are drawn from a standard deck of
cards. What is the probability they are all
jacks?
Let’s do a more complicated example before
getting to the next group of problems.
Example: A class consists of 10 girls and 9 boys.
If 5 of the students are to be selected at
random, determine the probability that 3 boys
and 2 girls are selected.
For the numerator, I need to select:
3 of 9 boys: This will give a 9C3 in the
numerator.
2 of 10 girls: This will give a 10C2 in the
numerator.
For the denominator, I can select:
5 of 19 club members: The denominator will be
19C5.
𝑃(3 𝑏𝑜𝑦𝑠 𝑎𝑛𝑑 2 𝑔𝑖𝑟𝑙𝑠 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑 ) =
(𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑜𝑢𝑝𝑠 𝑤𝑖𝑡ℎ 3 𝑜𝑓 9 𝑏𝑜𝑦𝑠)(𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑜𝑢𝑝𝑠 𝑤𝑖𝑡ℎ 2 𝑜𝑓 10 𝑔𝑖𝑟𝑙𝑠)
=
9𝐶3∗10𝐶2
19𝐶5
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑜𝑢𝑝𝑠 𝑜𝑓 5 𝑜𝑓 19 𝑐𝑙𝑢𝑏 𝑚𝑒𝑚𝑏𝑒𝑟𝑠)
84∗45
= 11628 = .3250
Answer: 32.5%
Homework #17-32 (Write your answer as a
percent rounded to 1 decimal place.)
17) A class consists of 19 girls and 15 boys. If
12 of the students are to be selected at
random, determine the probability that 4 boys
and 8 girls are selected.
18) A club has 8 boys and 5 girls. 3 members
are selected at random, determine the
probability that 2 boys and 1 girl is selected.
19) Of 35 people attending a dance 28 have a
college degree. If 4 people at the dance are
selected at random, find the probability 3 have
a college degree and 1 does not.
20) Of 50 people attending a concert 30 have
seen the group before. If 8 people at the
concert are selected at random, find the
probability that 6 have seen the group before
and 2 have not.
21) A class of 16 people contains 4 people
whose birthday is in October. If 3 people from
the class are selected at random, find the
probability that 2 have a birthday in October
and 1 doesn’t.
22) A class of 25 people contains 5 Gemini’s. If
4 people from the class are selected at random,
find the probability that 1 Gemini and 3 people
that are not Gemini’s are selected.
23) A committee of 4 is to be randomly
selected from a group of 7 teachers and 8
students. Find the probability the committee
will consist of 3 students and 1 teacher.
24) A committee of 6 is to be randomly
selected from a group of 7 teachers and 8
students. Find the probability the committee
will consist of 4 teachers and 2 students.
25) Anthony’s wallet contains 11 bills of the
following denominations, four $5 bills, five $10
bills, one $20 bill and one $50 bill. If he selects
five bills at random, determine the probability
that he selects 2 $5 bills and 3 tens.
26) Anthony’s wallet contains 10 bills of the
following denominations, four $5 bills, two $10
bills, three $20 bill and one $50 bill. If he
selects 3 bills at random, determine the
probability that he selects 2 tens and a five.
27) 3 cards are drawn from a standard deck of
cards. What is the probability they are two
hearts and 1 spade?
28) 4 cards are drawn from a standard deck of
cards. What is the probability they are 2 clubs
and 2 diamonds?
29) 2 cards are drawn from a standard deck of
cards. What is the probability they are 1 red
and 1 black?
30) 5 cards are drawn from a standard deck of
cards. What is the probability they are 3 black
and 2 red?
Answers: 1) 19C5 / 34C5 = .0417 4.2% 3)
28C4 / 35C4 = .391 = 39.1%
5) 4C3 / 16C3 = .007 = 0.7% 7) 8C4 / 15C4 =
.051 = 5.1% 9) 4C2 / 8C2 = .214 = 21.4%
11) 13C3 / 52C3 =.013 = 1.3% 13) 26C2 /
52C2 = .245 = 24.5%
15) 4C2 / 52C2 = .0045 = .5%
17) (15C4 * 19C8) / 34C12 = .188 = 18.8%
19) (28C3 * 7C1)/35C4 = .438 = 43.8%
21) (4C2 * 12C1) / 16C3 = .129 = 12.9%
23) (8C3 * 7C1) / 15C4 = .287 = 28.7%
25) (4C2 * 5C3) / 11C5 = .1298 = 13%
27) (13C2 * 13C1) / (52C3) = .046 = 4.6%
29) (26C1 * 26C1) / 52C2 = .5098 = 51%