The Binomial Distribution
Jayden makes $70\%$ of his free throws. In tonight's game he gets $5$ shots. How likely is it that he makes at least $4$ of them? Every shot is a yes/no result, and he repeats it a fixed number of times. That pattern shows up everywhere: a pollster asks $200$ people yes or no, a factory checks $50$ parts as good or defective, a student guesses on a multiple-choice quiz. The binomial distribution models the number of "successes" in situations like these. By the end of this page you will be able to answer Jayden's question.
Probability distributions
A random variable $X$ assigns a number to each outcome of a random process. It is discrete if its possible values can be listed (0, 1, 2, 3, …). A probability distribution lists every value of $X$ with its probability, usually in a table.
Why add up to 1? The table lists every possible outcome, and exactly one of them happens each time, so together they account for 100% of the probability.
In a popular strategy board game, a player rolls a special four-sided (tetrahedral) die numbered from 1 to 4. The random variable $X$ is defined as the "score obtained…
In a popular strategy board game, a player rolls a special four-sided (tetrahedral) die numbered from 1 to 4. The random variable $X$ is defined as the "score obtained on the die".
- a) Create the probability distribution table for the variable $X$.
- b) Calculate the expected value or mean $\mu$ and interpret its meaning in the context of the game.
A café near the high school has analyzed the number of coffees each customer orders in a morning. The probability distribution of the variable $X$ = "number of coffees…
A café near the high school has analyzed the number of coffees each customer orders in a morning. The probability distribution of the variable $X$ = "number of coffees per customer" is given by the following table:
| $x_i$ | 1 | 2 | 3 |
|---|---|---|---|
| $P(X = x_i)$ | $0,65$ | $0,25$ | $k$ |
- a) Find the value of $k$ so that it is a valid probability distribution.
- b) What is the probability that a customer orders at least 2 coffees?
Bernoulli trials and the binomial setting
A Bernoulli trial has only two outcomes: success (probability $p$) or failure (probability $q = 1 - p$). "Success" just means the outcome we are counting, even if it is bad news, like a defective part.
A quick checklist (some teachers call it BINS): Binary outcomes (success/failure), Independent trials, a fixed Number of trials, the Same probability $p$ on every trial. All four must hold.
When you are asked to define the distribution, say what $X$ counts and give $n$ and $p$. For example: "Let $X$ = number of negative messages among $6$ messages. Then $X \sim B(6, 0.10)$."
Determine logically whether the following random experiments can be modelled by a binomial distribution. If so, state their parameters $n$ and $p$: a) Toss a coin 15…
Determine logically whether the following random experiments can be modelled by a binomial distribution. If so, state their parameters $n$ and $p$:
- a) Toss a coin 15 times and record the number of heads obtained.
- b) Draw, one by one and without replacement, 3 cards from a standard deck of cards and count the number of diamonds.
- c) Ask 40 people chosen at random in a shopping center if they have seen the latest superhero movie, knowing that $30\%$ of the population has already seen it.
It is known that $10\%$ of the messages an influencer* receives on their social media accounts are negative comments (the rest are positive or neutral comments). If they…
It is known that $10\%$ of the messages an influencer receives on their social media accounts are negative comments (the rest are positive or neutral comments). If they have received 6 new messages today:
- a) Define the random variable and its distribution.
- b) Calculate the probability that exactly 2 of those messages are negative comments.
The binomial formula
How do we find $P(X = k)$? Think about Jayden's $5$ shots ($p = 0.7$, $q = 0.3$) and exactly $4$ makes. One way is to make the first four and miss the last. Because the shots are independent, we multiply: $0.7 \cdot 0.7 \cdot 0.7 \cdot 0.7 \cdot 0.3 = 0.7^4 \cdot 0.3$. But the miss could be any of the $5$ shots, and every one of those orders has the same probability.
Why? Each specific order with $k$ successes and $n - k$ failures has probability $p^k q^{n-k}$ (multiplying independent events), and there are $\binom{n}{k}$ such orders. Some problems call this formula the probability function of $X$. On a TI-84, $\binom{n}{k}$ is nCr (MATH → PRB), and binompdf(n, p, k) gives $P(X = k)$ directly. Use the formula to show your work, and the calculator to check.
In an electronic components factory, the probability of a circuit being defective is $0,02$. If they are packed in boxes of 5 units, calculate the probability that there…
In an electronic components factory, the probability of a circuit being defective is $0,02$. If they are packed in boxes of 5 units, calculate the probability that there are no defective circuits in a box.
Given a binomial distribution $X \sim B(4; 0,3)$, construct its complete probability distribution table by calculating the probabilities for all possible values of the…
Given a binomial distribution $X \sim B(4; 0,3)$, construct its complete probability distribution table by calculating the probabilities for all possible values of the variable ($k = 0, 1, 2, 3, 4$). Show that the sum of all the probabilities is equal to 1.
In a sports center, $75 \%$ of the users enrolled in swimming classes are able to swim 100 feet continuously without resting. If we choose 6 users at random: a) What is…
In a sports center, $75 \%$ of the users enrolled in swimming classes are able to swim 100 feet continuously without resting. If we choose 6 users at random:
- a) What is the probability that all of them are able to swim that distance?
- b) What is the probability that none of them are able to?
- c) Calculate the probability that half of them (exactly 3) meet the requirement.
The whole distribution: table and graph
Computing $P(X = k)$ for every $k = 0, 1, \dots, n$ gives the full distribution. Here is $X \sim B(5, 0.4)$, with $q = 0.6$.
| $k$ | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| $P(X = k)$ | $0.07776$ | $0.2592$ | $0.3456$ | $0.2304$ | $0.0768$ | $0.01024$ |
For example, $P(X = 2) = \binom{5}{2}(0.4)^2(0.6)^3 = 10 \cdot 0.16 \cdot 0.216 = 0.3456$. The six probabilities add up to $1$, as they must. A bar graph makes the shape easy to see.
Notice the peak sits near $n \cdot p = 5(0.4) = 2$. We will use that idea in a moment.
In a game of a well-known board game, a player needs to roll at least one "6" when rolling a fair six-sided die in 5 attempts to avoid being eliminated. Let $X$ be the…
In a game of a well-known board game, a player needs to roll at least one "6" when rolling a fair six-sided die in 5 attempts to avoid being eliminated. Let $X$ be the random variable that counts the number of times the number "6" is rolled in the 5 rolls.
- a) Determine the probability function of the variable $X$.
- b) Calculate the probability that the player is not eliminated (that is, they roll at least one six).
- c) Represent the distribution graphically using a bar chart.
Mean and standard deviation
You do not need to compute $\sum x \cdot P(x)$ term by term for a binomial.
Why? A binomial count is the sum of $n$ independent Bernoulli trials. Each has mean $p$ and variance $pq$, and for independent trials means and variances add. The mean $\mu = np$ also matches intuition: if $30\%$ of $200$ people agree, you expect about $60$.
Sometimes the problem works backward: you are given $\mu$ and $\sigma^2$ and must find $n$ and $p$.
An archer has a probability of hitting the target of $80\%$. If they shoot a round of 10 independent arrows, calculate the parameters of the corresponding binomial…
An archer has a probability of hitting the target of $80\%$. If they shoot a round of 10 independent arrows, calculate the parameters of the corresponding binomial distribution:
- a) Expected number of hits (mean $\mu$).
- b) Standard deviation $\sigma$.
An email server detects that $10 \%$ of the messages it receives are "spam" (junk mail). The security filter analyses the messages in blocks of 20. a) Find the expected…
An email server detects that $10 \%$ of the messages it receives are "spam" (junk mail). The security filter analyses the messages in blocks of 20.
- a) Find the expected number of "spam" emails per block and its standard deviation.
- b) Calculate the probability that there is exactly one "spam" message in a block.
- c) If the block is marked as "suspicious" when it contains more than 3 "spam" messages, what is the probability that a block is marked as such?
On an assembly line for e-sports components, it has been determined that the probability of a microchip being defective is $p$. It is known that in a batch of $n$…
On an assembly line for e-sports components, it has been determined that the probability of a microchip being defective is $p$. It is known that in a batch of $n$ microchips, the mean number of defective components is $12$ and the variance is $10,56$.
- a) Calculate the values of the parameters $n$ (batch size) and $p$ (probability of a defect).
- b) If a random sample of $10$ microchips is drawn from a new batch, what is the probability that there are exactly $2$ defective ones? And the probability that there is at most one?
"At least one," "at most," and "more than"
Real questions ask for ranges: "at most $2$ vans," "more than $6$ customers," "at least one defect." Adding many terms is slow, so use the complement.
Why? "No successes" means all $n$ independent trials failed, which has probability $q^n$. Also, "at most $k$" and "more than $k$" cannot both happen and together cover every outcome, so their probabilities add to $1$. Choose whichever side has fewer terms. On a TI-84, binomcdf(n, p, k) gives $P(X \le k)$.
A multiple-choice exam consists of 8 questions with 4 options each, of which only one is correct. If a student answers all the questions at random: a) What is the…
A multiple-choice exam consists of 8 questions with 4 options each, of which only one is correct. If a student answers all the questions at random:
- a) What is the probability that they do not answer any question correctly?
- b) Using the complementary event, calculate the probability that they answer at least one question correctly.
A courier company uses electric vans for its urban deliveries. It is known from experience that the probability of a van needing an additional recharge before finishing…
A courier company uses electric vans for its urban deliveries. It is known from experience that the probability of a van needing an additional recharge before finishing its shift is $0,15$. If the company has a fleet of 8 vans operating simultaneously:
- a) Calculate the probability that, on a given day, at most 2 vans need the additional recharge.
- b) What is the probability that more than 6 vans complete their shift without needing an extra recharge? (Be careful to correctly define what you consider a "success").
It has been observed that $5 \%$ of the LED bulbs of a certain brand leave the factory defective. A hotel buys boxes of 15 units to renew the lighting in its corridors.…
It has been observed that $5 \%$ of the LED bulbs of a certain brand leave the factory defective. A hotel buys boxes of 15 units to renew the lighting in its corridors.
- a) What is the probability that there are no defective bulbs in a box?
- b) If the hotel returns the box if it finds more than one defective bulb, what is the probability that a box is returned?
- c) If 100 boxes are bought, in how many of them is it expected to find exactly 2 defective bulbs? (Use the probability calculated for a single box).
Typical values: $\mu \pm \sigma$
The standard deviation tells you what counts as "typical." For $X \sim B(20, 0.5)$ (heads in $20$ coin flips), $\mu = 10$ and $\sigma = \sqrt{20(0.5)(0.5)} = \sqrt 5 \approx 2.24$. The interval $[\mu - \sigma, \mu + \sigma]$ is about $[7.76, 12.24]$. Only whole numbers can occur, so the values inside are $8, 9, 10, 11, 12$.
Adding the five shaded probabilities gives about $0.74$. So roughly three-quarters of the time, the count lands within one standard deviation of the mean.
A recent study on digital habits states that $35 \%$ of American teenagers aged 15 to 16 prefer to use podcasts as their main source of entertainment over conventional…
A recent study on digital habits states that $35 \%$ of American teenagers aged 15 to 16 prefer to use podcasts as their main source of entertainment over conventional social networks. If we select a random group of 10 students in 11th grade:
- a) Justify why this experiment can be modeled by a binomial distribution and define its parameters $n$ and $p$.
- b) Calculate the probability that exactly 4 of them prefer podcasts.
- c) Find the mean and the standard deviation of the number of students who prefer podcasts in this group.
A chain of "vintage" clothing stores has found that $20 \%$ of the customers who enter their establishment make a purchase of more than $\$50…
A chain of "vintage" clothing stores has found that $20 \%$ of the customers who enter their establishment make a purchase of more than $\$50$. If 12 customers enter independently in one morning:
- a) What is the probability that the number of customers who spend more than $\$50$ is equal to the mean of the distribution?
- b) Calculate the probability that at least 3 customers make such a purchase.
An exam consists of $80$ multiple-choice questions with $4$ answer options each, of which only one is correct. A student decides to answer $12$ questions they do not…
An exam consists of $80$ multiple-choice questions with $4$ answer options each, of which only one is correct. A student decides to answer $12$ questions they do not know how to answer at random. Let $X$ be the variable that counts the number of correct answers out of those $12$.
- a) Calculate the interval $[\mu - \sigma, \mu + \sigma]$ for this variable.
- b) Determine the probability that the number of correct answers lies outside this interval. Comment on whether the result obtained is consistent with the expected dispersion in a distribution of this type.
The most likely value (mode)
The mode of a binomial distribution is the value of $k$ with the largest probability, the tallest bar. To find it, compare each bar with the one before it:
$$\frac{P(X = k)}{P(X = k-1)} = \frac{(n - k + 1)\,p}{k\,q}$$This ratio is at least $1$ exactly when $(n - k + 1)p \ge k(1 - p)$, which simplifies to $k \le (n + 1)p$. So the bars keep rising while $k \le (n+1)p$ and fall afterward.
A multiple-choice exam consists of 10 questions with 4 answer options each, of which only one is correct. A student who has not studied at all decides to answer all the…
A multiple-choice exam consists of 10 questions with 4 answer options each, of which only one is correct. A student who has not studied at all decides to answer all the questions at random.
- a) Determine the probability that the student answers exactly 5 questions correctly.
- b) To pass the exam, it is necessary to answer at least 5 questions correctly. What is the probability that the student passes purely by chance?
- c) What is the most probable score (mode) that the student will obtain?
A streaming platform knows that $25\%$ of its users cancel their subscription before the third month. In a group of $8$ friends who have subscribed simultaneously: a)…
A streaming platform knows that $25\%$ of its users cancel their subscription before the third month. In a group of $8$ friends who have subscribed simultaneously:
- a) Calculate the probability that the number of friends who cancel their subscription is greater than the mean plus one standard deviation.
- b) If we represent by $P(X=k)$ the probability that exactly $k$ friends cancel, justify logically, without calculating all the values, what the value of $k$ is that maximizes this probability.
Finding unknown $n$ or $p$
Hard problems give clues about the distribution instead of $n$ and $p$. Translate each clue into an equation, then solve. Three clues come up again and again.
Clue 1: all failures or all successes. $P(X = 0) = q^n$ and $P(X = n) = p^n$. Rewrite the given numbers as powers with the same base and compare exponents. Remember that $n$ must be a whole number, which is a handy check.
Clue 2: one probability is a multiple of another. Write both with the formula, then cancel what they share.
Clue 3: "at least one with probability greater than…" This leads to an exponential inequality that needs logarithms.
The graph shows $P(\text{at least one success}) = 1 - q^n$ as $n$ grows. Slide $b$ (that is $q$) and see how quickly it climbs toward $1$.
A content creator on a well-known social network knows that the probability of a follower sharing one of their videos is exactly $p = 0,02$. If the creator publishes a…
A content creator on a well-known social network knows that the probability of a follower sharing one of their videos is exactly $p = 0,02$. If the creator publishes a series of videos and wants to be sure, with a probability greater than $95\%$, that at least one of their videos is shared by a specific follower who watches all their content:
- a) Determine the minimum number of videos, $n$, that they must publish to achieve this goal.
- b) If they finally publish $n = 150$ videos, calculate the expected value and the standard deviation of the number of videos shared by that follower. Interpret the meaning of the expected value in this context.
A courier company uses drones to deliver parcels in rural areas. The probability of a drone successfully completing a flight is $p$. The company has observed that the…
A courier company uses drones to deliver parcels in rural areas. The probability of a drone successfully completing a flight is $p$. The company has observed that the probability that, on a day with $4$ scheduled flights, all of them are successful is exactly twice the probability that only $3$ of them are.
- a) Calculate the value of $p$.
- b) With the value of $p$ obtained, if $6$ flights are scheduled one day, what is the probability that at least $2$ flights fail?
For a discrete random variable $X$ that follows a binomial distribution $B(n, p)$, it is known that the probability of obtaining zero successes is $P(X=0) = 0,0001$ and…
For a discrete random variable $X$ that follows a binomial distribution $B(n, p)$, it is known that the probability of obtaining zero successes is $P(X=0) = 0,0001$ and the probability of obtaining $n$ successes is $P(X=n) = 0,01$.
- a) Using the properties of powers, find the value of $n$ and that of $p$.
- b) Calculate the probability of obtaining a number of successes equal to the mode of the distribution. Hint: The mode in a binomial distribution is the integer value $k$ such that $(n+1)p - 1 \le k \le (n+1)p$.
Combining binomial events
Two more tools finish the toolkit.
Two independent variables. If $X$ and $Y$ come from independent processes, then $P(X = i \text{ and } Y = j) = P(X = i)\cdot P(Y = j)$. To find the probability of a combined event, list the pairs $(i, j)$ that make it happen, multiply for each pair, and add.
Conditional probability. The probability of $A$ given that $B$ has occurred is $P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)}$. When $A$ is a part of $B$ (for example $X \ge 2$ inside $X \ge 1$), then "$A$ and $B$" is just $A$, so $P(A \mid B) = \dfrac{P(A)}{P(B)}$.
Two e-sports players, Alex and Bea, compete in a target-shooting minigame. Alex has a probability of hitting the target of $70\%$, while Bea has a probability of $60\%$.…
Two e-sports players, Alex and Bea, compete in a target-shooting minigame. Alex has a probability of hitting the target of $70\%$, while Bea has a probability of $60\%$. If each of them takes $5$ independent shots:
- a) What is the probability that Alex gets more hits than Bea?
- b) What is the probability that between the two of them they get exactly $9$ hits?
A six-sided die is rolled $n$ times. Let $X$ be the random variable that expresses the number of times a multiple of $3$ is obtained. If it is known that the probability…
A six-sided die is rolled $n$ times. Let $X$ be the random variable that expresses the number of times a multiple of $3$ is obtained. If it is known that the probability of not obtaining any multiple of $3$ in the $n$ rolls is $16/81$:
- a) Calculate the number of rolls $n$.
- b) For that value of $n$, calculate the probability of obtaining a multiple of $3$ at least twice, given that it has appeared at least once.