SME
Grade 10 Math · Problem type
Probability with combinations
Use this for “3 students are picked at random from 5 girls and 4 boys. What is the chance all 3 are girls? Exactly 2?”
Diagram
P
=
C(5,
3)
×
C(4, 0)
C(9, 3)
=
10
84
=
5
42
≈
0.119:
the
groups
that
count
over
all
the
groups
3 places in order: 9 × 8 × 7 = 504 orders.
Each
group
of
3
comes
in
3!
=
6
orders:
504
6
=
84
groups.
This is an example. Type your own values to start a new problem.
C(
,
)
×
C(
,
−
)
C(
+
,
)
=
5
42
P
n
Everyone
worked out (not typed)
f
Favorable groups
worked out (not typed)
t
Groups in all
worked out (not typed)
Formulas
n
=
a
+
b
9 = 5 + 4
f
=
C(
a
,
k
)
×
C(
b
,
r
−
k
)
10 = C(5, 3) × C(4, 3 − 3)
f
=
C(
b
,
r
)
when
k
is
0
10 = C(4, 3) when 3 is 0
f
=
C(
a
,
r
)
when
k
is
r
10 = C(5, 3) when 3 is 3
t
=
C(
n
,
r
)
84 = C(9, 3)
P
=
f
t
5
42
=
10
84
Assumptions
•
Every group of r is equally likely, so count groups with combinations.
•
Favorable groups: C(a, k) ways to pick k from the first group, times C(b, r − k) for the rest.
•
All groups: C(a + b, r). The probability is the first over the second.
Step-by-step
WE KNOW
a = 5, b = 4, r = 3, k = 3
FIND
everyone (n), favorable groups (f), groups in all (t), probability (P)
Step 1 · Find everyone (n)
Use n = a + b
Everyone in both groups.
n
=
5
+
4
n
=
9
Step 2 · Find favorable groups (f)
Use f = C(a, k) × C(b, r − k)
Pick
k
from
the
first
group
and
the
other
r
−
k
from
the
second:
multiply
the
ways.
f
=
C(5,
3)
×
C(4,
3
−
3)
f
=
10
Step 3 · Find groups in all (t)
Use t = C(n, r)
The
ways
to
choose
any
r
from
everyone.
t
=
C(9,
3)
t
=
84
Step 4 · Find probability (P)
Use P = f ÷ t
Favorable groups over all the equally likely groups.
P
=
10
84
P
=
5
42
Check
Put the numbers back in. Both sides should match.
9 = 5 + 4 ✓
10 = C(5, 3) × C(4, 3 − 3) ✓
10 = C(5, 3) × C(4, 3 − 3) ✓
10 = C(5, 3) ✓
84 = C(9, 3) ✓
5
42
=
10
84
✓
← Previous lesson
Combinations: order doesn’t matter
Next lesson →
Conditional probability and independence
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Related lessons
Sample spaces, the addition rule, permutations and combinations
Main lesson
Mutually exclusive events
Use this for “P(red) = 0.25 and P(blue) = 0.40 for one marble. Find P(red or blue).”
The complement rule
Use this for “The chance of rain is 0.35. What is the chance of no rain?”
Neither event
Use this for “Of 40 students, 22 play soccer, 15 basketball and 8 both. How many play neither, and what is the chance of neither?”
Listing equally likely outcomes
Use this for “Three coins are tossed. What is the chance of exactly two heads?”
Permutations: order matters
Use this for “In how many ways can 8 runners take gold, silver and bronze?”
Combinations: order doesn’t matter
Use this for “In how many ways can a committee of 3 be chosen from 8 people?”