SME
Grade 11 Math · Algebra 2
Probability distributions: binomial distributions and expected value
Use this for “Each of 10 trials succeeds with chance 0.3. What is the chance of exactly 2 successes, and how many are expected?”
Chart
Binomial bars: n = 10 trials, each a success with chance p = 0.3.
E = n × p = 10 × 0.3 = 3
σ = √(n × p × (1 − p)) = 1.4491
The chance of exactly 2: 0.2335.
Orders of k successes, C(n, k): 45
Probability of exactly k successes, P(X = k): 0.2335
This is an example. Type your own values to start a new problem.
P(X
=
k
)
=
C(
n
,
k
)
×
p
×
(1
−
)
−
=
P
C
Orders of k successes, C(n, k)
worked out (not typed)
E
Expected successes E(X)
calculated
σ
Standard deviation
worked out (not typed)
Formulas
C
=
C
(
n
,
k
)
45
=
C
(10,
2)
P
=
C
×
p
k
×
(1 −
p
)
n
−
k
0.2335
=
45
×
0.3
2
×
(1 − 0.3)
10 − 2
E
=
n
×
p
3 = 10 × 0.3
σ
=
√(
n
×
p
×
(1
−
p
))
1.4491 = √(10 × 0.3 × (1 − 0.3))
Assumptions
•
There are n trials, and each one is a success or not.
•
Every trial has the same chance p, and the trials are independent.
•
E is the long-run average count of successes, not a promise for one round of n trials.
Step-by-step
WE KNOW
n = 10, p = 0.3, k = 2
FIND
orders of k successes, C(n, k) (C), probability of exactly k successes, P(X = k) (P), expected successes E(X) (E), standard deviation (σ)
Step 1 · Find orders of k successes, C(n, k): C
Use C = C(n, k)
The
ways
to
pick
which
k
of
the
n
trials
are
the
successes.
C
=
C
(10,
2)
C
=
45
Step 2 · Find probability of exactly k successes, P(X = k): P
Use P = C × pᵏ × (1 − p)ⁿ⁻ᵏ
Each
order
has
k
successes
at
p
and
n
−
k
failures
at
1
−
p
.
P
=
45
×
0.3
2
×
(1 − 0.3)
10 − 2
P
=
45
×
0.09
×
0.7
8
P
=
45
×
0.09
×
0.05765
P
=
0.2335
Step 3 · Find expected successes E(X): E
Use E = n × p
On
average,
p
of
the
n
trials
succeed.
E
=
10
×
0.3
E
=
3
Step 4 · Find standard deviation (σ)
Use σ = √(n × p × (1 − p))
The typical distance of the count from its mean.
σ
=
√(10
×
0.3
×
(1
−
0.3))
σ
=
√
10 × 0.3 × 0.7
σ
=
√
3 × 0.7
σ
=
√
2.1
σ
=
1.4491
Check
Put the numbers back in. Both sides should match.
45
=
C
(10,
2)
✓
0.2335
≈
45
×
0.3
2
×
(1 − 0.3)
10 − 2
✓
3 = 10 × 0.3 ✓
1.4491 ≈ √(10 × 0.3 × (1 − 0.3)) ✓
Topic check
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