Exercises

Two fair dice are rolled. What is the probability that their total is 7? Give it to four decimals.

Of 150 patients with a condition, 81 have a fever, 54 feel fatigue and 27 have both. What is the probability that a patient has neither symptom?

A fair coin is flipped 5 times. What is the probability of at least one head?

Of 150 patients, 81 have a fever, 54 feel fatigue and 27 have both. Among the patients without a fever, what is the probability of fatigue?

A condition affects 2 % of the people screened. The test is positive for 90 % of the people with the condition and negative for 95 % of those without it. What is the probability that someone who tests positive has the condition?

Two machines fail on a given day with probabilities 0.1 and 0.05, independently of each other. What is the probability that at least one of them fails?

A café sells 0, 1, 2 or 3 cakes in its last hour, with probabilities 0.1, 0.3, 0.4 and 0.2. What is the expected number of cakes sold?

For the same café (0, 1, 2 or 3 cakes with probabilities 0.1, 0.3, 0.4 and 0.2), what is the standard deviation of the number of cakes sold?

For the same café, what is the smallest number of cakes to keep for the last hour so that the demand is met with a probability of at least 0.75?

A bookseller sells between 90 and 100 books a day, every total equally likely. What is the probability of selling between 92 and 97 books, both included?

Fixing a bug takes anywhere from 1 to 5 hours, every time equally likely. What is the probability that a fix takes less than 2.2 hours?

Fixing a bug takes anywhere from 1 to 5 hours, every time equally likely. How long do the quickest 75 % of fixes take at most (the 75th percentile, in hours)?

A bookseller sells between 90 and 100 books a day, every whole number equally likely. An analyst reports the chance of selling at most 95 books as uniform.cdf(95, loc=90, scale=10) = 0.5. What is the right probability?

Fixes of a small bug take anywhere from 1 to 1.5 hours, every time equally likely. A chart labels the flat line at height 2 as "probability", and a reader concludes that every fix time has probability 2. What is the probability that a fix takes between 1.1 and 1.3 hours?

Historically, 5 % of a company's LED bulbs are faulty. In a random sample of 10 bulbs, what is the probability that none is faulty?

With the same fault rate of 5 %, what is the probability that three or more of the 10 bulbs are faulty?

A basketball player scores 93 % of free throws and takes two, independently. What is the probability of scoring at least one?

In a large company, 10 % of sales trainees are rated outstanding. For 10 trainees chosen at random, what is the probability that exactly two are rated outstanding?

With 10 % of trainees rated outstanding, what is the probability that two or more of 10 trainees are rated outstanding?

Of all sales trainees, 10 % are rated outstanding and 75 % excellent. For 10 trainees chosen at random, what is the probability that exactly eight are rated either outstanding or excellent?

A dose is unsatisfactory with probability one in three. For a batch of 10 doses, an analyst enters p = 0.33 and reports the chance of at most 3 unsatisfactory doses as 0.5684. What is the right probability?

A help desk receives on average 18 calls an hour. What is the probability that no call arrives in a given 10-minute period?

A help desk receives on average 4 calls an hour. What is the probability of 5 or more calls in an hour?

A report has on average 0.8 typing errors a page. What is the probability that a page has at most one error?

Each of 1,000 card payments is fraudulent with probability 0.002, independently. Using the Poisson approximation, what is the probability that exactly 3 are fraudulent?

Monthly phone bills are normal with a mean of 850 rupees and a standard deviation of 150 rupees. What is the probability that a bill is more than 1,200 rupees?

With the same bills (normal, mean 850 rupees, standard deviation 150 rupees), above what amount lie the top 15 % of bills? Give it in rupees to four decimals.

A machine fills bottles with a normal amount, mean 500 ml and standard deviation 20 ml. What is the probability that a bottle holds between 470 and 525 ml?

Two-litre bottles hold a normal amount of drink, mean 2.0 litres and standard deviation 0.05 litres. At least what amount do 99 % of bottles contain? Give it in litres to four decimals.

The simulated share's daily log returns have a mean of 0.00123 and a standard deviation of 0.0279, so the yearly log return (250 trading days) is modelled as normal with mean 250 × 0.00123 and standard deviation √250 × 0.0279. An analyst reports the chance of a drop of more than 50 % in a year as norm.cdf(-0.5, 250*0.00123, np.sqrt(250)*0.0279) = 0.034. What is the right probability?

With the simulated share's daily log returns modelled as normal with mean 0.00123 and standard deviation 0.0279, an analyst computes the chance of a fall of more than 5 % in a day as norm.cdf(-0.05, 0.00123, 0.0279). What is the right probability?

Deliveries take on average 35 minutes with a standard deviation of 8 minutes, roughly normally. What share of deliveries take longer than 47 minutes?

For the same deliveries (normal, mean 35 minutes, standard deviation 8 minutes), what is the exact probability that a delivery takes between 27 and 51 minutes?

An e-mail goes to 200 customers, each opening it with probability 0.25, independently. Using the normal approximation with the continuity correction, what is the probability of at least 60 opens?

A packing line fills bags to a mean of 500 g with a standard deviation of 4 g: that is the process. Five bags weighed this morning have a mean of 503 g and a standard deviation of 2 g. To judge a 508 g bag, an analyst computes z = (508 − 503)/2 = 2.5. What is the right z-score?

Customers arrive at a café at 12 an hour, at random. What is the probability that the next customer arrives within 3 minutes?

A component's lifetime is exponential with a mean of 2,000 hours. By what time have half the components failed (the median lifetime, in hours)?

The same component (exponential lifetime, mean 2,000 hours) has already run for 3,000 hours. What is the probability that it runs at least another 500 hours?

A maker's batteries last on average 60 months, with a standard deviation of 6 months, roughly normally. What is the probability that the mean lifetime of 40 batteries chosen at random is below 58 months?

Deliveries take on average 35 minutes, with a standard deviation of 8 minutes. What is the probability that the mean of 16 deliveries lies between 33 and 37 minutes?

Fixing a bug takes anywhere from 1 to 5 hours, every time equally likely. A sprint closes 40 bugs. Using the CLT, what is the probability that their average fix time is more than 3.3 hours?

Order values vary with a standard deviation of £18. How many orders must be averaged for the standard error of the mean to be at most £1.5?

A share's daily log returns have a standard deviation of 0.018. If the days are independent, what is the standard deviation of its log return over a year of 250 trading days?

Waiting times at a counter are exponential with a mean of 2 minutes. For the mean of 30 waits an analyst uses the CLT, \(\bar{X} \approx N(2, \tfrac{2}{\sqrt{30}})\), and reports \(P(\bar{X} > 2.8) = 0.0142\), since "n = 30 is large enough". The mean of 30 exponential waits in fact follows a gamma distribution, gamma(a=30, scale=2/30). What is the exact probability?

A random sample of 40 households used on average 310 kWh in a quarter. A large earlier study found a standard deviation of 89 kWh, taken as known. What is the lower end of the 95 % confidence interval for the mean usage, in kWh to four decimals?

A 99 % confidence interval is built from a sample of 12 values with σ unknown. What is the critical value t*?

Ten wildfires caused damage with a mean of 53.6 and a standard deviation of 40.24 (thousands of dollars). What is the upper end of the 90 % t interval for the mean damage, to four decimals?

Caffeine per cup has a known standard deviation of 7 mg. What is the smallest number of cups for which a 95 % z interval for the mean has a margin of error of at most 1 mg?

Four measurements are 4, 7, 9 and 12. What is their sample standard deviation s, computed with n − 1?

From 50 cups, a 95 % confidence interval for a machine's mean caffeine runs from 111.24 to 115.12 mg (σ = 7 mg). A manager reads it as "95 % of cups contain between 111.24 and 115.12 mg". If cups are normal with mean 113.18 mg and standard deviation 7 mg, what share of cups really lies between 111.24 and 115.12 mg?