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Quizzes > Mathematics > Advanced Math

Master Binomial Distribution with Our Practice Test

Tackle binomial distribution questions and practice problems - test your skills now!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
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Ready to master discrete probability? Our Ultimate Binomial Distribution Test is a free, scored quiz built to sharpen your binomial distribution test skills and challenge you with authentic binomial distribution questions. You'll tackle scenarios that mirror real data, honing your grasp of combinations and success/failure trials while answering questions on binomial probability distribution. Need a warm-up? Try our binomial theorem practice problems or review core concepts with probability practice questions . Whether you're a student, data enthusiast, or professional, this interactive test of binomial distribution practice problems and focused binomial questions will boost your confidence. Hit "Start Quiz" now and take the leap!

What is the probability mass function for observing k successes in n independent Bernoulli trials with success probability p?
[n!/(k!(n-k)!)] p^k
C(n,k) p^k (1-p)^(n-k)
[n!/(k!(n-k)!)] (p/(1-p))^k
p^k (1-p)^(n-k)
The binomial formula is given by the combination C(n,k)=n!/(k!(n-k)!) multiplied by p^k (1-p)^(n-k), which accounts for k successes and n?k failures in any order. This ensures the probabilities across all possible k from 0 to n sum to 1. It arises from selecting which trials are successes and the probability of each sequence. Binomial Distribution
For a Binomial(10, 0.5) distribution, what is P(X = 0)?
0.005
0.00097656
0.01
0.001
P(X=0) = C(10,0)(0.5)^0(0.5)^10 = 1×1×(0.5)^10 ? 0.00097656. Because there are no successes, only the failure term remains. This tiny probability reflects all 10 trials failing. Binomial Distribution Details
What is the mean of a Binomial(n, p) distribution?
np(1-p)
n(1-p)
np
?[np(1-p)]
The expected value of a Binomial(n,p) is np, reflecting the average number of successes over n trials each with probability p. It follows from summing individual Bernoulli expectations. This linearity property is fundamental to binomial models. Mean of Binomial
What is the variance of a Binomial(n, p) distribution?
np(1-p)
n^2p(1-p)
np
?[np(1-p)]
Variance for Binomial(n,p) is np(1-p), derived from summing the variances of n independent Bernoulli trials, each with variance p(1-p). It measures the spread around the mean np. This formula is key to understanding binomial variability. Variance of Binomial
Which of the following is NOT an assumption of the binomial distribution?
There are only two possible outcomes per trial
Probability of success varies from trial to trial
The number of trials n is fixed
Each trial is independent
A binomial model assumes fixed n, two outcomes per trial, constant success probability p, and independent trials. Varying p would break the identical distribution requirement. Thus changing success probability disqualifies the binomial structure. Binomial Assumptions
For X ~ Binomial(n, p), how is P(X > k) most directly computed?
?_{i=k+1}^n C(n,i)p^i(1-p)^(n-i)
1 - ?_{i=0}^k p^i(1-p)^(n-i)
?_{i=0}^k C(n,i)p^i(1-p)^(n-i)
1 - ?_{i=k+1}^n C(n,i)p^i(1-p)^(n-i)
P(X>k) is the sum of probabilities from k+1 up to n: ?_{i=k+1}^n C(n,i)p^i(1-p)^(n-i). This directly adds all outcomes exceeding k successes. Alternatively it equals 1 minus the cumulative up to k. Binomial Distribution Sum
If the success probability p = 0 in a Binomial(n, p), what is the distribution of X?
Degenerate at 0 successes
Uniform on 0 to n
Normal with mean 0
Poisson with ?=0
When p=0 every trial always fails, so X=0 with probability 1. This is a degenerate distribution concentrated at 0. No variability remains since no successes can occur. Degenerate Distribution
If the success probability p = 1 in a Binomial(n, p), what is the distribution of X?
Normal with mean n
Degenerate at n successes
Poisson with ?=n
Uniform on 0 to n
When p=1 each trial always succeeds, so X=n with probability 1. This degenerate distribution puts all mass at n. There is zero chance of fewer successes. Degenerate Distribution
A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads?
0.2500
0.3125
0.6250
0.5000
P(X=3)=C(5,3)(0.5)^5=10×1/32=0.3125. There are 10 ways to choose which flips are heads and each sequence has probability (0.5)^5. This direct calculation uses the binomial formula. Binomial Calculator
For X ~ Binomial(20, 0.2), what is P(X ? 2) approximately?
0.500
0.800
0.069
0.214
P(X?2)=?_{i=0}^2 C(20,i)(0.2)^i(0.8)^(20?i) ?0.0115+0.0576+0.1368=0.214. Summing the first three probabilities gives the cumulative value. Calculators or tables often assist with these sums. Binomial CDF
If X ~ Binomial(15, 0.3), what is P(X = 5)?
0.324
0.014
0.206
0.102
P(X=5)=C(15,5)(0.3)^5(0.7)^10 ?3003×0.00243×0.02825?0.206. Use combination and powers for exact probability, then numeric multiplication. Many statistical tools yield the same result. Binomial Distribution
In 12 independent trials with success probability 0.2, what is the expected number of successes?
4.8
2.0
2.4
3.6
The mean of Binomial(n,p) is np = 12×0.2 = 2.4. This is the average number of successes over many repetitions. It follows directly from the definition of the expectation. Mean of Binomial
What is the standard deviation of a Binomial(50, 0.4) distribution?
5.000
4.900
3.464
2.345
Standard deviation = ?[np(1-p)] = ?[50×0.4×0.6] = ?12 ?3.464. It quantifies the typical deviation from the mean. The formula comes from summing Bernoulli variances. Variance of Binomial
Which scenario CANNOT be modeled by a binomial distribution?
Drawing 5 cards from a deck without replacement and counting aces
Testing 20 independent components with failure probability p
Rolling a die 8 times and counting sixes
Flipping a fair coin 10 times and counting heads
Binomial trials require constant p and independent trials. Drawing cards without replacement changes the probability each draw, violating independence and identical distribution. Replacement or independence is key for binomial models. Binomial Conditions
If X ~ Binomial(n?,p) and Y ~ Binomial(n?,p) are independent, what is the distribution of X+Y?
Binomial(max(n?,n?), p)
Normal(np, np(1-p))
Binomial(n?+n?, p)
Poisson(np)
The sum of independent Binomial(n?,p) and Binomial(n?,p) is Binomial(n?+n?,p). Each trial remains Bernoulli(p) and counts combine. This property follows from adding independent counts. Additivity
What is the mode of a Binomial(n, p) distribution when (n+1)p is not an integer?
?(n+1)p?
?(n+1)p?
np
(n+1)p
When (n+1)p is not integer, the unique mode is the floor of (n+1)p. If (n+1)p is integer, there are two modes. This result comes from comparing successive probabilities. Mode of Binomial
Using normal approximation, Binomial(100, 0.2) ? N(?,?²). What are ? and ?²?
?=80, ?²=16
?=50, ?²=20
?=20, ?²=16
?=20, ?²=4
For Binomial(n,p), mean ?=np=100×0.2=20 and variance ?²=np(1-p)=100×0.2×0.8=16. These parameters define the approximating normal. Large n and p not extreme justify the approximation. Normal Approximation
Which expression uses continuity correction to approximate P(15 ? X ? 25) for X ~ Binomial(100,0.2)?
?((25.5-20)/16) ? ?((14.5-20)/16)
?((25-20)/4) ? ?((15-20)/4)
?((25-20)/16) ? ?((15-20)/16)
?((25.5-20)/4) ? ?((14.5-20)/4)
Continuity correction adds ±0.5: P(15?X?25)??((25.5??)/?)??((14.5??)/?) with ?=20, ?=4. This adjusts discrete-to-continuous approximation. It improves accuracy for moderate n. Continuity Correction
When approximating Binomial(50,0.02) by a Poisson distribution, what is the Poisson parameter ??
50
2
0.02
1
Poisson approximation for Binomial(n,p) uses ?=np=50×0.02=1 when n is large and p small. This simplifies calculations for rare events. Many statistical tables use this practical rule. Poisson Approximation
What is the moment generating function (MGF) of a Binomial(n, p) distribution?
(1-p + pt)^n
np(1-p)t
e^{np t}
[(1-p) + p e^t]^n
The MGF M_X(t)=E[e^{tX}]=[(1-p)+p e^t]^n since each trial contributes a factor (1-p + p e^t). Raising to n accounts for independence. This compact form is used to derive moments. MGF Definition
If X ~ Binomial(n, p), then Y = n ? X follows which distribution?
Binomial(n, p)
Normal(np, np(1-p))
Poisson(np)
Binomial(n, 1-p)
If X counts successes, then n?X counts failures. Each trial has failure probability 1?p, so Y ~ Binomial(n,1?p). The complement of each Bernoulli(p) is Bernoulli(1-p). Binomial Complement
What is the skewness of a Binomial(n, p) distribution?
(1?2p)/?[np(1?p)]
?[np(1?p)]
np(1?p)(1?2p)
0
Skewness = (1?2p)/?[np(1?p)], describing asymmetry. A p<0.5 yields positive skew, p>0.5 negative skew. It derives from the third standardized moment. Skewness of Binomial
Given observed X successes in n trials, what is the maximum likelihood estimator (MLE) of p for a Binomial(n, p)?
?(X/n)
(X+1)/(n+2)
X/(n+1)
X/n
The likelihood L(p)?p^X(1?p)^{n?X}. Differentiating log-likelihood and setting to zero gives X/p ? (n?X)/(1?p)=0, so p?=X/n. This is the MLE. MLE for Binomial
Which canonical parameter identifies the binomial distribution as a member of the exponential family?
? = np
? = ln(p/(1-p))
? = 1-p
? = p
In exponential family form, the natural (canonical) parameter for Binomial(n,p) is ? = ln(p/(1-p)). This transforms the likelihood into exp{?x ? n ln(1+e^?)}. Recognizing this parameterization classifies it within the one-parameter exponential family. Exponential Family
Which theorem describes the convergence of the standardized Binomial distribution to the normal distribution as n???
De Moivre - Laplace theorem
Law of Large Numbers
Chebyshev's Inequality
Central Limit Theorem
The De Moivre - Laplace theorem is a special case of the Central Limit Theorem for binomial distributions, stating that (X?np)/?[np(1?p)] approaches N(0,1) as n??. It historically predates the general CLT. This result underpins normal approximations for binomial probabilities. De Moivre - Laplace
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Study Outcomes

  1. Understand Binomial Distribution Fundamentals -

    Gain a clear grasp of the binomial distribution parameters - number of trials, success probability, and success count - to confidently approach binomial distribution test problems.

  2. Calculate Binomial Probabilities -

    Master the process of computing probabilities for specified numbers of successes using the binomial probability distribution formula and apply it to diverse practice problems.

  3. Apply Formulas to Binomial Distribution Practice Problems -

    Skillfully use combinations and probability rules to solve binomial distribution practice problems and reinforce your computational accuracy.

  4. Interpret Results from Binomial Distribution Questions -

    Develop the ability to analyze probability outcomes and draw meaningful conclusions from modeled scenarios on your binomial distribution questions.

  5. Evaluate Real-World Scenarios Using Questions on Binomial Probability Distribution -

    Translate real-world situations into binomial probability distribution questions and determine the likelihood of different outcomes with confidence.

  6. Optimize Performance with Scored Binomial Distribution Test Feedback -

    Leverage immediate scoring and feedback to identify strengths and target areas for improvement in your binomial distribution test proficiency.

Cheat Sheet

  1. Binomial Distribution Fundamentals -

    Recall that a binomial distribution models the number of successes in n independent trials with constant probability p (Ross, 2014). Use the mnemonic "BINS" (Binary outcomes, Independent trials, Number fixed, Success probability constant) to ensure all conditions are met before applying P(X=k).

  2. Probability Mass Function (PMF) -

    Apply the formula P(X=k) = C(n,k)·p^k·(1−p)^(n−k), where C(n,k)=n!/(k!(n−k)!) gives the number of ways to get k successes (Grinstead & Snell, 1997). Practice expanding small factorial examples, e.g., C(5,2)=10, to build fluency in computing probabilities by hand.

  3. Mean and Variance -

    Know that the expected value μ = n·p and variance σ² = n·p·(1−p) (Hogg & Tanis, 2015). A quick check: if p=0.3 and n=20, then μ=6 and σ²=4.2, helping you gauge distribution spread before deep calculation.

  4. Cumulative Probabilities and Tables -

    For P(X ≤ k), sum P(X=0) through P(X=k) or use statistical software/graphs from resources like Wolfram Alpha or R's pbinom() function. Visualizing with cumulative tables (e.g., from Khan Academy) speeds up answers for "at most" or "at least" questions.

  5. Real-World Applications -

    Binomial models appear in quality control (defective items), clinical trials (success/failure), and survey sampling. Practice designing your own scenario, such as "What's the chance exactly 3 out of 10 patients respond?" to cement concept-to-context mapping.

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