Probability Calculator

Compute simple probabilities, binomial probabilities, and normal distribution values using experiment-friendly inputs.

Basic Probability

Number of Successes

Total Outcomes

Result -

Binomial Probability

Number of Trials (n)

Number of Successes (k)

Probability of Success (p)

Result -

Normal Distribution

X Value

Mean

Standard Deviation

Result -

Probability: Basics, Binomial, and Normal Distribution

Probability measures how likely events are to happen, and models like the binomial and normal distributions help describe repeated and continuous outcomes.

This guide matches your other calculators with structured boxes, formulas, examples, tables, pitfalls, and practice problems focused on the three tools above.

What Is Probability?

Probability is a number between 0 and 1 (or 0% and 100%) that represents how likely an event is to occur.

Key terms

Outcome: A possible result of an experiment (e.g., rolling a 4).
Event: A set of outcomes (e.g., rolling an even number).
Sample space: The set of all possible outcomes.

For equally likely outcomes, the basic formula is:

Probability = (Number of favorable outcomes) / (Total outcomes)

Example:
Rolling a fair die, probability of getting an even number (2, 4, 6):
Favorable = 3, total = 6 → P(even) = 3/6 = 1/2 = 0.5 = 50%.

Experiment	Event	Probability
Coin flip	Heads	1/2
Die roll	Number > 4	2/6 = 1/3
Deck of cards	Drawing a heart	13/52 = 1/4

Basic Probability Calculator

The “Basic Probability” card uses the ratio of successes to total outcomes to compute probability and its percentage.

Formula used

P(Event) = successes / total outcomes

Example 1:
5 successes out of 20 trials.
P = 5/20 = 0.25 = 25%.

Example 2:
A class has 30 students, 12 of whom are wearing glasses. If one student is chosen at random, P(glasses) = 12/30 = 0.4 = 40%.

Successes	Total	Probability	Percentage
3	10	0.3	30%
8	40	0.2	20%
15	50	0.3	30%

Binomial Probability

The binomial model describes the probability of getting exactly k successes in n independent trials, each with success probability p.

Conditions

Fixed number of trials n.
Each trial has two outcomes (success/failure).
Probability of success p is the same for each trial.
Trials are independent.

Formula: P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ

Example 1:
Flip a fair coin 5 times (n = 5, p = 0.5). What is P(exactly 2 heads)?
C(5, 2) = 10, so P = 10 · (0.5)² · (0.5)³ = 10 · (0.5)⁵ = 10/32 = 0.3125 = 31.25%.

Example 2:
A machine produces items with success rate p = 0.8. In n = 4 items, P(exactly 3 good) is:
C(4, 3) = 4; P = 4 · (0.8)³ · (0.2)¹ = 4 · 0.512 · 0.2 = 0.4096 (≈ 40.96%).

n	k	p	P(X = k)
5	0	0.5	(0.5)⁵ = 0.03125
5	5	0.5	(0.5)⁵ = 0.03125
4	2	0.3	C(4,2)·0.3²·0.7²

Normal Distribution and the Calculator

The normal (Gaussian) distribution is a bell‑shaped curve defined by its mean and standard deviation, often used to model continuous data like heights or test scores.

Standardization and CDF

Z‑score: z = (x − μ) / σ
The calculator uses an approximation of the normal CDF P(X ≤ x) via the error function erf.

Example 1:
Test scores are normal with mean μ = 70 and σ = 10. What is P(X ≤ 85)?
z = (85 − 70)/10 = 1.5. The calculator approximates P(X ≤ 85) ≈ 0.933 (93.3%).

Example 2:
Heights are normal with μ = 170 cm, σ = 8 cm. For x = 162, z = (162 − 170)/8 = −1. The calculator returns P(X ≤ 162) ≈ 0.159 (15.9%).

z	P(Z ≤ z) (approx)	Comment
0	0.5	Center of the distribution
1	0.8413	About 84.1%
−1	0.1587	About 15.9%
2	0.9772	About 97.7%

Where These Probability Models Are Used

Basic, binomial, and normal probabilities appear in statistics, quality control, finance, and everyday decisions.

🎲 Games and Randomness

Basic probability models dice, cards, lotteries, and random events in board and video games.

📦 Quality Control

Binomial probabilities estimate defect rates and the chance that a sample contains a certain number of faulty items.

📈 Finance and Risk

Normal distributions approximate returns and measurement noise when modeling prices and indices.

🧪 Experiments and Surveys

Probabilities summarize success rates, confidence levels, and expected outcomes in experiments and polls.

📊 Standardized Tests

Scores are often modeled as normal, so percentiles correspond to areas under the normal curve.

🤖 Machine Learning

Probabilistic models power classifiers, regression, and Bayesian inference in modern AI systems.

Common Probability Mistakes

Probability problems often go wrong because of misinterpreted events, wrong denominators, or mixing models.

❌ MISTAKE 1: Using the wrong total outcomes
Forgetting to count all possible outcomes leads to incorrect probabilities.
✅ Carefully define the sample space before computing P = successes / total.

❌ MISTAKE 2: Treating dependent events as independent
Drawing cards without replacement changes probabilities each draw.
✅ Use binomial only when trials are independent with constant p.

❌ MISTAKE 3: Misreading normal distribution tails
Confusing P(X ≤ x) with P(X ≥ x) gives inverted answers.
✅ Remember the calculator computes P(X ≤ x); for ≥ use 1 − P(X ≤ x).

❌ MISTAKE 4: Invalid probability values
Using p < 0 or p > 1 in binomial models is impossible.
✅ Ensure probabilities lie between 0 and 1 inclusive.

Practice Problems with Solutions

Try these and verify your answers using the three calculator cards above.

Basic Probability

Problem 1: A bag has 4 red, 5 blue, and 1 green ball. If one ball is drawn at random, find P(blue).

Show Solution

Total = 4 + 5 + 1 = 10; blue = 5 → P(blue) = 5/10 = 0.5 = 50%.

Problem 2: In 40 questions, you answered 32 correctly. What is the probability of a correct answer? (Use Basic Probability.)

Show Solution

P(correct) = 32/40 = 0.8 = 80%.

Binomial Probability

Problem 3: A coin is tossed 6 times. What is P(exactly 4 heads)? (n = 6, k = 4, p = 0.5.)

Show Solution

C(6,4) = 15; P = 15·(0.5)⁴·(0.5)² = 15·(0.5)⁶ = 15/64 ≈ 0.2344 (23.44%).

Problem 4: A basketball player makes each free throw with probability p = 0.7. In 5 shots, what is P(exactly 3 made)?

Show Solution

C(5,3) = 10; P = 10·0.7³·0.3² = 10·0.343·0.09 ≈ 0.3087 (30.87%).

Normal Distribution

Problem 5: Heights in a class are normal with μ = 160 cm, σ = 6 cm. Find P(X ≤ 172).

Show Solution

z = (172 − 160)/6 = 12/6 = 2. Use the calculator with x=172, mean=160, std=6 to get P(X ≤ 172) ≈ 0.9772 (97.72%).

Problem 6: Test scores follow N(μ = 75, σ = 8). What proportion of students score less than 67?

Show Solution

z = (67 − 75)/8 = −8/8 = −1. Use the calculator with x=67, mean=75, std=8 to get P(X ≤ 67) ≈ 0.1587 (15.87%).