How to Calculate Binomial Probability (With Examples)

Have you just been introduced to the binomial probability formula and are wondering how you can use it to calculate binomial probability? This article shows you how to apply the formula to find exact and cumulative binomial probabilities using examples.

Recall: The binomial probability formula is P(X=x) = ⁿC_{x ·}p^x(1 − p)^n−x

Where:

P(X=x) is the probability we want to compute
ⁿC_x is read as n “combination” x and is calculated using the formula [n!/x!(n−x)]!
p is the probability of success
1-p is the probability of failure
x is the total number of successful events

Solving Binomial Probability Problems (With Examples)

In binomial probability questions, you may be asked to compute exact or cumulative probabilities. While finding exact probabilities is straightforward, cumulative probabilities require students to understand key terms such as “less than,” “at most,” “greater than,” or “at least.”

The following examples are tailored to help you master how to compute any binomial probability question.

Example 1. Exact Probability, P(X=x)

A random process has a probability of success of 0.3 on each trial. If the process is repeated 6 times, what is the probability of obtaining exactly 2 successes?

Solution

This is a binomial probability problem, where:

Probability of success, p = 0.3
The number of independent trials, n = 6

We need to find P(x=2)

In this case, we only need to apply the binomial formula and plug in the known values

That is:

P(x=2) = ⁶C_{2 ·}0.3²(1 − 0.3)⁶⁻²

= 15*0.3²*(0.7)⁴

=15×0.09×0.2401

≈0.3241

Therefore, P(X = 2) = 0.3241

Example 2. Less Than, P(X<x)

A factory finds that 20% of its products are defective. If 5 products are randomly selected, what is the probability that fewer than 2 products are defective?

Solution

This is a binomial probability problem, where:

Probability of success, p =0.20
The number of independent trials, n = 5

Fewer than 2 means that we need to find P(x<2)

As you know, the binomial formula allows us to compute exact probabilities.

Since the binomial distribution is a discrete probability, we can find P(x<2) as follows:

P(x<2) = P(x=0) + P(x=1)

In other words, we need to find the exact probabilities of x = 0 and x = 1 and sum them

P(x= 0) = ⁵C_{0 ·}0.2⁰(1 − 0.2)⁵⁻⁰

=1 * 1 *0.8⁵

= 0.3277

P(x=1) = ⁵C_{1 ·}0.2¹(1 − 0.2)⁵⁻¹

= 5*0.2*0.8⁴

=0.4096

Therefore, P(x<2) = 0.3277+0.4096

=0.7373

Example 3. Less Than or Equal To, P(X≤x)

A student guesses answers on 4 multiple-choice questions, each with one correct answer out of four. What is the probability that the student answers at most 1 question correctly?

Solution

This is a binomial probability question, where:

Probability of success, p = 1/4 = 0.25
The number of independent trials, n = 4

Since the question requires us to find at most 1, we need to find P(X≤1)

Using the discrete probability concept,

P(X≤1) = P(X=0) + P(X=1)

P(x=0) = ⁴C_{0 ·}0.25⁰(1 − 0.25)⁴⁻⁰

=1*1*0.75⁴

=0.3164

P(x=1) = ⁴C_{1 ·}0.25¹(1 − 0.25)⁴⁻¹

=4 * 0.25 * 0.75³

=0.4219

Therefore, P(X≤1) = 0.3164 + 0.4219

=0.7383

Example 4. Greater Than, P(X>x)

A basketball player makes a free throw with probability 0.6. If the player attempts 5 free throws, what is the probability that the player makes more than 3 shots?

Solution

This is a binomial probability problem, where:

Probability of success, p = 0.6
The number of independent trials, n = 5

Since the question says “More than 3 shots,” we need to find P(X>3)

Applying the discrete distribution concept of binomial probability,

P(X>3) = P(x=4) + P(x=5)

P(x=4) = ⁵C_{4 ·}0.6⁴(1 − 0.6)⁵⁻⁴

= 5 * 0.1296 *0.4¹

=0.2592

P(x=5) = ⁵C_{5 ·}0.6⁵(1 − 0.6)⁵⁻⁵

=1 * 0.07776 * 1

=0.0778

Therefore, P(X>3) = 0.2592 + 0.0778

=0.3370

Example 5. Greater Than or Equal To, P(X≥x)

A call center agent successfully resolves 70% of customer issues. If the agent handles 6 calls, what is the probability that at least 4 issues are resolved?

Solution

This is a binomial probability problem, where:

Probability of success, p = 0.70
The number of independent trials, n = 6

Since the question says “at least 4 issues,” we need to find P(X≥4)

Using the discrete probability concept,

P(X≥4) = P(X=4) + P(X=5) + P(X=6)

P(x=4) = ⁶C_{4 ·}0.7⁴(1 − 0.7)⁶⁻⁴

=15 * 0.2401 * 0.3²

= 0.3241

P(x=5) = ⁶C_{5 ·}0.7⁵(1 − 0.7)⁶⁻⁵

=6 * 0.16807 * 0.3¹

=0.3025

P(X=6) = ⁶C_{6 ·}0.7⁶(1 − 0.7)⁶⁻⁶

=1 * 0.1176 *0.3⁰

=0.1176

Therefore, P(X≥4) = 0.3241 + 0.3025 + 0.1176

=0.7442

Want a quick way to find binomial probabilities? Try our binomial probability calculator. The calculator computes both exact and cumulative probabilities once you specify the probability of success (p), the number of trials (n), and the probability of interest (x).

Practice Questions

PQ1. A biased coin has a probability of heads equal to 0.3. If the coin is tossed 5 times, find P(X=2).

A. 0.1323
B. 0.3087
C. 0.3602
D. 0.5282

PQ2. A machine produces items with a defect rate of 10%. If 6 items are selected, find P(X<2).

A. 0.1143
B. 0.4686
C. 0.8857
D. 0.5314

PQ3. A student answers 3 true/false questions by guessing. Find P(X≤1).

A. 0.125
B. 0.375
C. 0.500
D. 0.875

PQ4. A quality inspector checks 4 products, each with a probability of defect of 0.2. Find P(X>1).

A. 0.1808
B. 0.2624
C. 0.5904
D. 0.8192

PQ5. A researcher finds that 60% of participants respond positively to a treatment. If 5 participants are selected, find P(X≥3).

A. 0.3174
B. 0.6826
C. 0.4518
D. 0.8640

Answers