The **binomial probability distribution calculator** is designed to compute the **probability** of specific outcomes in a series of **independent Bernoulli trials**.

**Bernoulli **trials are experiments that yield only two possible outcomes, typically classified as **success** or **failure**, with the probability of success remaining constant throughout the trials.

The **binomial probability distribution** is a **discrete probability distribution** that quantifies the number of successes in a fixed number of independent Bernoulli trials.

The **binomial probability distribution calculator** requires the following inputs:

**Number of trials (n)**: The total number of independent Bernoulli trials.**Probability of success (p)**: The probability of success in each trial.**Number of successes (k)**: The number of successes for which you want to calculate the probability.

The calculator then outputs the probability of observing

ksuccesses out ofntrials, given the probability of successpin each trial.

## Binomial Probability Distribution Calculation Chart

Trials (n) | Probability of Success (p) | Number of Successes (k) | Probability (P(X=k)) |
---|---|---|---|

5 | 0.3 | 1 | 0.3888 |

5 | 0.3 | 2 | 0.3456 |

5 | 0.3 | 3 | 0.1728 |

5 | 0.3 | 4 | 0.0486 |

5 | 0.3 | 5 | 0.0054 |

10 | 0.5 | 3 | 0.1172 |

10 | 0.5 | 4 | 0.2344 |

10 | 0.5 | 5 | 0.2344 |

10 | 0.5 | 6 | 0.1172 |

10 | 0.5 | 7 | 0.0312 |

## Binomial Probability Distribution Formula

The formula is:

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

Where:

P(X=k)is the probability of observingksuccesses inntrialsnis the number of trialskis the number of successespis the probability of success in each trialnCkis the binomial coefficient, representing the number of ways to chooseksuccesses out ofntrials.

Consider an example where you have a **fair coin** (with a **50% probability** of getting heads) and you flip it **5 times**. What is the probability of getting exactly **3 heads**?

To calculate this, we can use the binomial probability distribution formula:

**P(X=3) = 5C3 ***(0.5)^3* (0.5)^(5-3)

**P(X=3) = 10 ***0.125* 0.25

**P(X=3) = 0.3125**

The probability of getting exactly **3 heads** in **5 coin flips** is **0.3125** or **31.25%**.

## How do you calculate the probability of a binomial distribution?

To calculate the probability of a **binomial distribution**, utilize the binomial probability distribution formula:

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

Where:

P(X=k)is the probability of observingksuccesses inntrialsnis the number of trialskis the number of successespis the probability of success in each trialnCkis the binomial coefficient, representing the number of ways to chooseksuccesses out ofntrials.

Suppose you roll a **6-sided die** **10 times** and want to calculate the probability of getting exactly **4 sixes**.

In this scenario:

n(the number of trials) =10k(the number of successes) =4p(the probability of success in each trial) =1/6(since the probability of rolling a six on a fair die is1/6)

**Plugging these values into the formula:**

**P(X=4) = 10C4 ***(1/6)^4* (5/6)^(10-4)

**P(X=4) = 210 ***(1/1296)* (625/4096)

**P(X=4) = 0.0405 or 4.05%**

The probability of rolling exactly **4 sixes** in **10 rolls** of a fair die is approximately **4.05%**.