The binomial distribution table is a table that shows probabilities associated with the binomial distribution. To use the binomial distribution table, you only need three values:
- n: the number of trials
- r: the number of “successes” during n trials
- p: the probability of success on a given trial
Using these three numbers, you can use the binomial distribution table to find the probability of obtaining exactly r successes during n trials when the probability of success on each trial is p.
The following examples illustrate how to read the binomial distribution table.
Example 1
Question: Jessica makes 60% of her free-throw attempts. If she shoots 6 free throws, what is the probability that she makes exactly 4?
To answer this question, we can look up the value in the binomial distribution table that corresponds to n = 6, r = 4, and p = 0.60:
The probability that Jessica makes exactly 4 out of 6 free throws is 0.311.
Example 2
Question: Jessica makes 60% of her free-throw attempts. If she shoots 6 free throws, what is the probability that she makes less than 4?
To find this probability, we actually have to add up the following probabilities:
P(makes less than 4) = P(makes 0) + P(makes 1) + P(makes 2) + P(makes 3)
So, we can look up each of these four probabilities in the binomial distribution table and add them up:
According to the table, P(makes less than 4) = .004 + .037 + .138 + .276 = 0.455.
The probability that Jessica makes less than 4 free throws is 0.455.
Example 3
Question: Jessica makes 60% of her free-throw attempts. If she shoots 6 free throws, what is the probability that she makes 4 or more?
To find this probability, we have to add up the following probabilities:
P(makes 4 or more) = P(makes 4) + P(makes 5) + P(makes 6)
So, we can look up each of these three probabilities in the binomial distribution table and add them up:
According to the table, P(makes 4 or more) = .311 + .187 + .047 = 0.545.
The probability that Jessica makes 4 or more free throws is 0.545.
Thank you for the explanation 🙂
Thanks for explanation
Explain and more and relative exponential distribution.
Thank for your teaching about binomial distribution but I also need help concerning question of cumulative binomial distribution.
You’re very welcome! I’m glad the explanation of the **binomial distribution** helped. Now let’s dive into the **cumulative binomial distribution** — it’s just the next step!
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### 🧠 **Quick Recap: What Is Binomial Distribution?**
The **binomial distribution** gives the probability of getting exactly **k successes** in **n independent trials**, each with success probability **p**.
The formula is:
\[
P(X = k) = \binom{n}{k} p^k (1 – p)^{n-k}
\]
—
### 📈 What Is Cumulative Binomial Distribution?
The **cumulative** binomial distribution calculates the **probability of getting *at most* k successes**, that is:
\[
P(X \leq k) = P(X = 0) + P(X = 1) + \cdots + P(X = k)
\]
This is very useful when you want the **total probability up to a certain number of successes**.
—
### ✅ Example:
Suppose:
– \( n = 5 \) trials
– \( p = 0.6 \) chance of success in each trial
– You want to find:
**What is the probability of getting *at most* 3 successes?**
So you compute:
\[
P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
\]
Each of these can be calculated using the binomial formula.
—
### 🔢 Using Python:
If you’re using Python with `scipy`, it becomes super easy:
“`python
from scipy.stats import binom
# Parameters
n = 5 # number of trials
p = 0.6 # probability of success
k = 3 # cumulative up to 3 successes
# Cumulative probability
prob = binom.cdf(k, n, p)
print(f”P(X ≤ {k}) = {prob:.4f}”)
“`
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