A binomial setting counts successes in a fixed number of independent trials, each with the same probability.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The four conditions
Binary outcomes, independent trials, a fixed number of trials, and the same probability throughout.
The probability
P(exactly k) = nCk · pᵏ · (1 − p)^(n − k). The combination counts which trials succeeded.
Mean and standard deviation
The mean is np and the standard deviation is √(np(1 − p)).
Geometric settings
A geometric variable counts trials until the first success. The number of trials is not fixed, which is the difference.
The geometric mean
The expected number of trials until the first success is 1 ÷ p. At p = 0.2 you expect 5 trials.
The binomial conditions
A fixed number of trials, two outcomes, constant probability, and independence. All four are required, and an exam answer using the binomial model is expected to verify them explicitly.
Mean and standard deviation
For a binomial count, the mean is np and the standard deviation √(np(1 − p)). Both are worth knowing, since they appear inside every proportion inference formula later in the course.
The geometric distribution
Same conditions, but the number of trials is not fixed — you count until the first success. Its mean is 1/p, so a probability of 0.2 means an average wait of five trials.
The 10% condition
Sampling without replacement breaks independence, but the effect is negligible if the sample is under 10% of the population. That condition is what lets the binomial model be used on real samples.
Step 2: Try It Yourself
Tap and try it out.
0 heads
1 head
2 heads
3 heads
2 heads has the most. It has 5 more than 0 heads.
Step 3: Watch an Example
One step at a time.
Watch Sana Check the Conditions
Sana draws 5 cards from a deck without replacement and counts the hearts.
- Step 1
The outcomes are binary, heart or not, so that condition holds.
Step 4: Your Turn
Practice makes it stick.
The Mean
Problem 1 of 2
n = 40 and p = 0.5. What is the mean number of successes?
The Wait
Problem 2 of 2
A geometric setting with p = 0.2. What is the expected number of trials until the first success?
Which Distribution
1 of 8
n = 100, p = 0.3. Mean number of successes?
2 of 8
n = 100, p = 0.5. Standard deviation?
3 of 8
Geometric with p = 0.25. Expected trials until the first success?
4 of 8
How many conditions must a binomial setting satisfy?
5 of 8
4 fair coin tosses. How many patterns have exactly 2 heads?
6 of 8
4 fair coin tosses. P(exactly 2 heads), to three decimal places?
7 of 8
Sort each setting by its distribution.
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8 of 8
n = 50, p = 0.2. Mean number of successes?
Step 5: Quick Check
Show what you know.
Question 1 of 2
n = 60 and p = 0.5. What is the mean number of successes?
Question 2 of 2
What separates a geometric setting from a binomial one?
What You Learned
- A binomial setting needs binary outcomes, independence, fixed trials and constant p.
- The mean is np and the standard deviation is √(np(1 − p)).
- A geometric variable counts trials until the first success, with mean 1 ÷ p.