Exercises: The binomial distribution

A new messaging app analysed that $30\,\%$ of sent messages contain at least one emoji. If a student sends a burst of $n = 6$ independent messages to a group, identify the parameters of the binomial distribution followed by the random variable $X$: "number of messages containing emojis" and write down the expression for its probability function $P(X = k)$.

In a first-person shooter video game, a player has a probability of $80\,\%$ of hitting the target each time they shoot. If they fire a series of $n = 4$ independent shots, calculate the exact probability of hitting the target on exactly $3$ occasions using the binomial distribution formula.

It is known that $15\,\%$ of subscribers to a streaming channel cancel their subscription before the end of the first month. If $100$ new subscribers are selected at random, calculate the expected number of cancellations (mean) and the standard deviation of this distribution.

A multiple-choice exam has $5$ questions, each with three answer options of which only one is correct. If a student answers all the questions at random, calculate the probability that they get none right. Represent the problem with a simplified tree diagram for the first two questions (indicate on each branch whether it is a correct answer or a miss and its probability).

Give reasons to state which of the following random variables follow a binomial distribution:

1. Drawing $4$ cards from a Spanish deck, one by one, without replacement, and counting the number of aces.

2. Rolling a fair die $10$ times and counting how many times an even number is obtained.

3. Asking $20$ people in a city for their favorite color from a list of $5$ colors.

On a photography social network, it is estimated that the probability of a follower "liking" a post by a given influencer is $p = 0.25$. If the influencer posts a story and $8$ followers view it independently, calculate the probability that it receives at least one "like".

Hint: Use the complementary event.

A wireless headphone manufacturer finds that $2\,\%$ of its devices leave the factory with a pairing defect. If they are packed in boxes of $10$ units, what is the probability that a box contains no defective headphones? Write only the formulation using combinations.

Given a discrete random variable that follows a $X \sim B(n=5, p=0.4)$ distribution, complete the following table of its probability mass function for the indicated values:

In a popular strategy video game, the probability that a "loot chest" contains a legendary skin is $15\%$. A player decides to open a pack of $8$ chests one after another. Assuming that the contents of each chest are independent of the others:

1. a) Identify the parameters of the binomial distribution followed by the random variable $X$: "number of legendary skins obtained".

1. b) Calculate the probability that the player obtains exactly $2$ legendary skins.

1. c) What is the probability that the player obtains no legendary skin in the whole pack?

An email server uses an advanced security filter that correctly detects $92\%$ of phishing emails. If $12$ malicious emails arrive in a company's inbox during one morning:

1. a) Calculate the probability that the filter detects at least $11$ of those emails.
2. b) Calculate the probability that the filter fails and lets through more than one malicious email.
3. c) On average, how many emails is the filter expected to detect? Calculate also the standard deviation of the distribution.

A college league basketball player has a free-throw success rate of $80\%$. During a crucial game, she is awarded $10$ free throws. Assume the throws are independent and that the probability of success is the same on each one.

1. a) Define the random variable and its distribution.

1. b) Calculate the probability that she makes exactly $8$ of the $10$ throws.

1. c) If she needs to make at least $9$ of the $10$ shots to win the game, what is the probability that her team wins thanks to her free throws?

In a study on sustainable mobility in a Mexican city, it was observed that $25\%$ of young people aged between $16$ and $18$ use an electric scooter to travel to high school. A random sample of $6$ students in this age group is selected.

1. a) Construct the probability mass function table for the variable $X$: "number of students who use an electric scooter".
2. b) What is the most likely number of successes (the mode of the distribution)?
3. c) Calculate the probability that half or more of the selected students use this mode of transportation.

An electronic component manufacturing company knows that the probability of a capacitor being defective is $5\%$. The capacitors are packed in boxes for distribution.

1. a) If a box contains $20$ capacitors, what is the probability of finding exactly one defective component?
2. b) What is the probability that in a box of $20$ units there are at least two defective ones?
3. c) What is the minimum number of capacitors a box should contain so that the probability of finding at least one defective one is greater than $0.9$?

It is estimated that in a certain region, $30\%$ of the population has blood group $O+$. If $7$ people are chosen at random for an emergency blood donation:

1. a) What is the probability that exactly $3$ of them are group $O+$?
2. b) What is the probability that none of the $7$ people are group $O+$?
3. c) If it is known that at least one of the people is group $O+$, what is the probability that exactly two are? (Hint: use the definition of conditional probability).

A multiple-choice exam consists of $15$ questions, each with $4$ possible answers of which only one is correct. A student who has not studied at all decides to answer all the questions at random.

1. a) What distribution does the variable $X$: "number of correct answers" follow? State its parameters.

1. b) If at least $8$ questions must be answered correctly to pass the exam, what is the probability that the student passes?

1. c) Calculate the expected value and explain its meaning in the context of the problem.

A pharmaceutical company claims that a new seasonal allergy treatment has an efficacy of $75\%$. The treatment is administered to a group of $9$ volunteer patients.

1. a) Calculate the probability that the treatment is effective in exactly $6$ patients.
2. b) What is the probability that the treatment fails in more than $2$ patients?
3. c) If the treatment is considered a commercial success if it works in at least $7$ of the $9$ patients, what is the probability of achieving such success?

A cloud data storage server uses an integrity verification system for each packet of information sent. It is known that the probability that a packet is transmitted successfully without errors is $p$. If 10 packets are sent independently, the probability that at least one of them arrives with errors is $0.65132$.

1. a) Determine the value of the probability $p$ of successful transmission of a packet.

1. b) If the security of the channel is increased and the error probability per packet is reduced to one quarter of the original, what is the minimum number of packets that must be sent so that the probability that all of them arrive without errors is less than $0.5$?

The figure shows, as a guide, how the probability that all packets arrive without errors decreases with the improved channel as the number of packets $n$ increases, and compares it with the threshold $0.5$. The exact value of $n$ must be obtained by calculation.

A drone component manufacturing company carries out strict quality control. It has been observed that the number of defective sensors in batches of 20 units follows a binomial distribution. If the standard deviation of this distribution is exactly $1.2$ sensors:

1. a) Calculate the possible values of the probability $p$ that a sensor is defective.

1. b) If it is known that the probability of a sensor being non-defective is greater than that of it being defective, calculate the probability that in a batch of 20 units there are exactly 2 defective sensors.

1. c) For the value of $p$ found in the previous section, if a technician checks the sensors one by one, what is the probability that the first defective sensor appears precisely on the fourth check?

An artificial intelligence algorithm classifies satellite images as "wooded area" or "non-wooded area". The algorithm has a probability $p$ of success. In a performance test with 15 images, it is known that the probability that the algorithm correctly classifies all of them is the same as the probability that it fails on all 15.

1. a) Justify, without performing complex calculations, what the value of $p$ must be.

1. b) Calculate the probability that the algorithm makes more than 12 errors in a series of 15 images.

1. c) If the developer claims that the expected number of correct classifications is 12, what would be the new value of $p$? With this new value, what is the probability that the algorithm correctly classifies at least 14 images?

On a microblogging social network, it is estimated that $15\%$ of the accounts are "bots". A researcher randomly selects a group of accounts for study.

1. a) What is the minimum size of the group of accounts that they must select to ensure, with a probability greater than $95\%$, that they will find at least one account that is a bot?

1. b) If the researcher finally decides to select a group of 12 accounts, what is the probability that the number of bots found lies within one standard deviation of the mean?

Five biased coins are tossed, each with probability of heads $p$ (the same for all coins, and the tosses are independent). It is known that the probability of getting 3 heads is exactly 10 times the probability of getting 5 heads.

1. a) Determine the value of $p$.

1. b) Calculate the probability of getting an even number of heads (including zero).

A communication system transmits bits ($0$ or $1$). Due to noise, each bit has a probability $\epsilon = 0.02$ of being received incorrectly. To improve reliability, a repetition code is used: each information bit is sent 5 times and the receiver decides the value of the bit by majority vote (if it receives three or more "1"s, it decides that a "1" was sent).

1. a) If a "1" is sent, what is the probability that the receiver incorrectly decides that a "0" was sent?

1. b) If a message of 10 information bits is transmitted (each protected by the 5-bit repetition code), calculate the probability that the complete message is received without errors.

1. c) Calculate the expected value and the variance of the number of information bits received correctly in the 10-bit message.

In an OLED screen manufacturing process, the probability that a screen has a dead pixel is $p$. The screens are packed in boxes of 10 units. It has been observed that $20\%$ of the boxes do not have any screen with dead pixels.

1. a) Calculate the value of $p$.

1. b) In a sample of 5 boxes, what is the probability that at least 2 boxes contain exactly one screen with dead pixels?

A delivery app guarantees delivery in less than 30 minutes. The probability that a delivery driver meets this guarantee is $p=0.9$. A delivery driver makes 20 deliveries in a day.

1. a) Calculate the probability that the number of on-time deliveries is greater than 17, given that at least 16 were delivered on time.

1. b) If the cost to the company for each late delivery is $100$ pesos and the profit for each on-time delivery is $40$ pesos, calculate the expected value of the total profit after the 20 deliveries.

1. c) Determine the probability that the total profit is positive.