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
| 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.
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