**Enter the values in binomial probability distribution calculator to compute probabilities in scenarios involving a fixed number of independent trials, each with two possible outcomes: success or failure. **

## Binomial Probability Distribution Calculator

This calculator helps you determine the binomial probability of getting exactly k successes in n trials.

The binomial probability is calculated using the formula: P(X = k) = C(n, k) * p^k * (1-p)^(n-k)

n (trials) | p (probability) | k (successes) | P(X = k) | P(X ≤ k) |
---|---|---|---|---|

10 | 0.5 | 5 | 0.2461 | 0.6230 |

20 | 0.3 | 6 | 0.2066 | 0.8670 |

15 | 0.7 | 10 | 0.1948 | 0.8479 |

30 | 0.4 | 12 | 0.1658 | 0.5535 |

50 | 0.6 | 35 | 0.0364 | 0.9780 |

**Explanation of calculations:**

- n = 10, p = 0.5, k = 5: This represents flipping a fair coin 10 times and getting exactly 5 heads.
- n = 20, p = 0.3, k = 6: This could represent a production line where each item has a 30% chance of being defective, and we’re calculating the probability of exactly 6 defective items in a batch of 20.
- n = 15, p = 0.7, k = 10: This might represent a basketball player with a 70% free throw percentage, calculating the probability of making exactly 10 out of 15 free throws.
- n = 30, p = 0.4, k = 12: This could represent a marketing campaign where each customer has a 40% chance of conversion, calculating the probability of exactly 12 conversions out of 30 customers.
- n = 50, p = 0.6, k = 35: This might represent a quality control scenario where each product has a 60% chance of passing inspection, calculating the probability of exactly 35 products passing out of 50.

**More Calculators : – Expected Value Calculator – Shannon Entropy Calculator**

## Binomial Distribution Probability Calculation Formula

The formula for calculating the probability of exactly k successes in n trials is:

**P(X = k) = C(n,k) * p^k * (1-p)^(n-k)**

Where:

**P(X = k)**is the probability of exactly k successes**C(n,k)**is the number of ways to choose k items from n items (combination)**p**is the probability of success on each trial**n**is the total number of trials**k**is the number of successes we’re interested in

The combination C(n,k) is calculated as:

**C(n,k) = n! / (k! * (n-k)!)**

Where **!** denotes the factorial operation.

## What is the cumulative probability of a binomial?

The **cumulative probability** of a **binomial distribution** refers to the probability of obtaining **up to and including** a certain number of successes.

It is often denoted as **P(X ≤ k)**, which means the probability of getting k or fewer successes in n trials.

To calculate the cumulative probability, we sum the individual probabilities for all outcomes from 0 to k:

**P(X ≤ k) = P(X = 0) + P(X = 1) + P(X = 2) + … + P(X = k)**

This calculation can be tedious for large values of n and k, which is why binomial probability distribution calculators are invaluable.

They can quickly compute both individual and cumulative probabilities for various combinations of n, p, and k.

## What is the probability of success in n trials?

The **probability of success in n trials** can have two interpretations:

**Probability of at least one success**: This is often what people mean when they ask about the probability of success in n trials. It’s calculated as the complement of the probability of no successes:**P(at least one success) = 1 – P(X = 0) = 1 – (1-p)^n****Expected number of successes**: This is the average number of successes you’d expect to see if you repeated the n trials many times. It’s calculated as:**E(X) = n * p**

For example, if you flip a fair coin (p = 0.5) 10 times (n = 10):

- The probability of getting at least one heads is:
**1 – (1-0.5)^10 ≈ 0.999**(99.9%) - The expected number of heads is:
**10 * 0.5 = 5**

## Binomial Distribution TI 84

The TI-84 is a popular graphing calculator that includes built-in functions for working with binomial distributions. Here’s how you can use the TI-84 for binomial distribution calculations:

**Binomial Probability Distribution Function (binompdf)**:- This function calculates P(X = k)
- Syntax:
`binompdf(n, p, k)`

- Access: Press [2nd] [DISTR], then select
`0:binompdf(`

- Example: To calculate the probability of exactly 5 successes in 10 trials with p = 0.5, enter
`binompdf(10, 0.5, 5)`

**Cumulative Binomial Distribution Function (binomcdf)**:- This function calculates P(X ≤ k)
- Syntax:
`binomcdf(n, p, k)`

- Access: Press [2nd] [DISTR], then select
`A:binomcdf(`

- Example: To calculate the probability of 5 or fewer successes in 10 trials with p = 0.5, enter
`binomcdf(10, 0.5, 5)`

**Graphing the Distribution**:- You can also graph the binomial distribution on the TI-84
- Enter the stat plot menu by pressing [2nd] [Y=]
- Select a plot and set it to histogram or bar graph
- Use
`binompdf(n, p, X)`

as the Xlist and leave Freq as 1 - Adjust your window settings and graph

Using the TI-84 for binomial distribution calculations can significantly speed up your work, especially for larger values of n and k. It’s particularly useful for students in statistics classes and professionals who need quick probability calculations in the field.