DP Mathematics HL Questionbank

• Syllabus

5.6

Sub sections and their related questions

Binomial distribution, its mean and variance.

• 12M.2.hl.TZ2.2a: the value of p ;
• 12M.2.hl.TZ2.2b: \({\text{P}}(X = 10)\) ;
• 12M.2.hl.TZ2.2c: \({\text{P}}(X \geqslant 15)\) .
• 12M.2.hl.TZ2.10c: The weights of the plums are normally distributed with a mean of 80 grams and a standard...
• 12N.3sp.hl.TZ0.1d: State the distribution of B .
• 12N.3sp.hl.TZ0.1e: Find P(B = 3) .
• 12N.3sp.hl.TZ0.1f: Find P(B = 5) .
• 08M.2.hl.TZ1.11: The lifts in the office buildings of a small city have occasional breakdowns. The breakdowns at...
• 08M.2.hl.TZ2.7: Over a one month period, Ava and Sven play a total of n games of tennis. The probability that...
• 08N.2.hl.TZ0.11Part A: (a)     A box of biscuits is considered to be underweight if it weighs less than 228 grams. It is...
• 11M.2.hl.TZ2.6b: John catches 6 fish. Calculate the probability that at least 4 of the fish weigh more than 1.4 kg.
• 09N.3sp.hl.TZ0.4: The random variable X has the distribution \({\text{B}}(n{\text{ , }}p)\) . (a)     (i)     Show...
• SPNone.2.hl.TZ0.12b: A random sample of ten of these birds is obtained. Let X denote the number of birds in the sample...
• 13M.1.hl.TZ1.13a: Show that the probability that Alfred wins exactly 4 of the games is \(\frac{{80}}{{243}}\).
• 13M.2.hl.TZ1.3b: Find the probability she is late at least once during the next week (Monday to Friday).
• 10M.2.hl.TZ1.12: Casualties arrive at an accident unit with a mean rate of one every 10 minutes. Assume that the...
• 10M.1.hl.TZ2.4: A biased coin is weighted such that the probability of obtaining a head is \(\frac{4}{7}\). The...
• 13M.2.hl.TZ2.11c: The random variable \(X \sim B(n,{\text{ }}p)\) has mean 4 and variance 3. (i)     Determine n...
• 11N.2.hl.TZ0.5a: find the probability that the 08:00 train is delayed exactly twice during any period of five work...
• 11N.2.hl.TZ0.5b: find the minimum number of work days for which the probability of the 08:00 train being delayed...
• 11M.2.hl.TZ1.12e: At 15:00 it is the end of the school day and it is assumed that the departure of the students...
• 09M.2.hl.TZ2.11: (a)     Show that the probability that a randomly selected bottle filled by Machine A contains...
• 14M.2.hl.TZ2.6: Six customers wait in a queue in a supermarket. A customer can choose to pay with cash or a...
• 15M.2.hl.TZ1.3a: State the distribution of \(W\), including the values of any parameters.
• 15M.2.hl.TZ1.3b: Write down the mean of \(W\).
• 15M.2.hl.TZ1.3c: Find \({\text{P}}(W > 89)\).
• 15M.2.hl.TZ2.10a: Farmer Suzie grows turnips and the weights of her turnips are normally distributed with a mean of...
• 15M.2.hl.TZ2.10b: Farmer Ray also grows turnips and the weights of his turnips are normally distributed with a mean...
• 15M.3sp.hl.TZ0.5a: Determine the probability generating function for \(X \sim {\text{B}}(1,{\text{ }}p)\).
• 15M.3sp.hl.TZ0.5b: Explain why the probability generating function for \({\text{B}}(n,{\text{ }}p)\) is a polynomial...
• 15M.3sp.hl.TZ0.5c: Two independent random variables \({X_1}\) and \({X_2}\) are such that...
• 15N.2.hl.TZ0.11c: There are elevators in the office building that take the office workers to their offices. Given...
• 16M.2.hl.TZ1.10: Students sign up at a desk for an activity during the course of an afternoon. The arrival of each...
• 16M.2.hl.TZ1.13c: Ten balls are taken from the bag. Find the probability that less than four of the balls are...
• 16M.2.hl.TZ1.13d: Find the least number of balls that must be taken from the bag for the probability of taking out...
• 16M.2.hl.TZ1.13e: Another bag also contains balls numbered 1 , 2 or 3. Eight balls are to be taken from this bag...
• 16M.1.hl.TZ2.5a: Find, in terms of \(p\), an expression for \({\text{P}}(X = 4)\).
• 16M.1.hl.TZ2.5b: (i)     Determine the value of \(p\) for which \({\text{P}}(X = 4)\) is a maximum. (ii)     For...
• 16M.2.hl.TZ2.6a: Find the probability that a randomly chosen glass sheet contains at least one flaw.
• 16M.2.hl.TZ2.6b: Find the expected profit, \(P\) dollars, per glass sheet.
• 16M.2.hl.TZ2.6c: Find the probability that it contains no flaws.
• 16N.2.hl.TZ0.3a: Show that...
• 16N.2.hl.TZ0.3b: Given that \({\text{P}}(X = 2) = 0.241667\) and \({\text{P}}(X = 3) = 0.112777\), use part (a) to...
• 17N.1.hl.TZ0.10b.i: Determine the mean of X.
• 17N.1.hl.TZ0.10b.ii: Determine the variance of X.
• 17N.2.hl.TZ0.6a: Find the probability that Lucca eats at least one banana in a particular day.
• 17N.2.hl.TZ0.6b: Find the expected number of weeks in the year in which Lucca eats no bananas.
• 18M.2.hl.TZ2.8a: Find the least possible value of n.
• 18M.2.hl.TZ2.8b: It is further given that P(X ≤ 1) = 0.09478 correct to 4 significant figures. Determine the...

Poisson distribution, its mean and variance.

• 12M.2.hl.TZ1.7: A fisherman notices that in any hour of fishing, he is equally likely to catch exactly two fish,...
• 12M.2.hl.TZ1.12a(i)(ii): Find the probability that in a certain week (Monday to Friday only) (i)     there are fewer than...
• 12M.2.hl.TZ1.12b(i)(ii): Due to the increased usage, it is found that the probability of more than 3 accidents in a day at...
• 12M.2.hl.TZ2.5a: the value of m ;
• 12M.2.hl.TZ2.5b: P (X > 2) .
• 12M.3sp.hl.TZ0.3a: Calculate the mean and variance of the number of injuries per week.
• 12M.3sp.hl.TZ0.3b: Explain why these values provide supporting evidence for using a Poisson distribution model.
• 12M.3sp.hl.TZ0.5a: show that \({m^2} = k(k + 1)\);
• 12M.3sp.hl.TZ0.5b: hence show that the mode of X is k .
• 12N.2.hl.TZ0.11a: If the museum is open 6 hours daily, find the expected number of visitors in 1 day.
• 12N.2.hl.TZ0.11b: Find the probability that the number of visitors arriving during an hour exceeds 100.
• 12N.2.hl.TZ0.11c: Find the probability that the number of visitors in each of the 6 hours the museum is open...
• 08M.2.hl.TZ1.11: The lifts in the office buildings of a small city have occasional breakdowns. The breakdowns at...
• 08M.2.hl.TZ2.11c: The number of telephone calls, T, received by Euler College each minute can be modelled by a...
• 08N.2.hl.TZ0.7: (a)     Ahmed is typing Section A of a mathematics examination paper. The number of mistakes that...
• 11M.2.hl.TZ2.12a: Find the probability that no accidents occur in a given month.
• 11M.2.hl.TZ2.12b: Find the probability that no accidents occur in a given 6 month period.
• 11M.2.hl.TZ2.12c: Find the length of time, in complete months, for which the probability that at least 1 accident...
• SPNone.2.hl.TZ0.12c: The number of eggs, Y , laid by female birds of this species during the nesting season is...
• 13M.2.hl.TZ1.10a: Find T, to the nearest minute, if \({\text{P}}(X \leqslant 3) = 0.6\).
• 13M.2.hl.TZ1.10b: It is now decided that the time between crossings, T, will be 10 minutes. The ferry can carry a...
• 10M.2.hl.TZ1.12: Casualties arrive at an accident unit with a mean rate of one every 10 minutes. Assume that the...
• 10M.2.hl.TZ2.6: The random variable X follows a Poisson distribution with mean \(\lambda \). (a)     Find...
• 10N.2.hl.TZ0.7: The random variable X follows a Poisson distribution with mean m and...
• 13M.2.hl.TZ2.9a: Find the probability that on a particular weekend, three requests are received on Saturday and...
• 11N.2.hl.TZ0.3a: Find the probability that no vehicles pass in a given minute.
• 11N.2.hl.TZ0.3b: Find the expected number of vehicles which pass in a given two minute period.
• 11N.2.hl.TZ0.3c: Find the probability that more than this expected number actually pass in a given two minute period.
• 11M.2.hl.TZ1.12d: At 15:00 it is the end of the school day and it is assumed that the departure of the students...
• 09M.2.hl.TZ1.4: (a) the probability that he catches at least one fish in the first hour; (b) the probability...
• 09M.2.hl.TZ2.11: (a)     Show that the probability that a randomly selected bottle filled by Machine A contains...
• 14M.3sp.hl.TZ0.1a: (i)     Find \({\text{P}}(X = 6)\). (ii)     Find \({\text{P}}(X = 6|5 \leqslant X \leqslant 8)\).
• 14M.2.hl.TZ1.9a: Find the probability that during a certain seven-day week, more than 40 birds have been seen on...
• 14M.2.hl.TZ2.8: The random variable \(X\) has a Poisson distribution with mean \(\mu \). Given that...
• 13N.2.hl.TZ0.11a: The number of cats visiting Helena’s garden each week follows a Poisson distribution with mean...
• 14N.2.hl.TZ0.11a: On a randomly chosen day, find the probability that (i)     there are no complaints; (ii)    ...
• 14N.2.hl.TZ0.11b: In a randomly chosen five-day week, find the probability that there are no complaints.
• 14N.2.hl.TZ0.11c: On a randomly chosen day, find the most likely number of complaints received. Justify your answer.
• 14N.2.hl.TZ0.11d: The department store introduces a new policy to improve customer service. The number of...
• 14N.3sp.hl.TZ0.4d: Could either \(S\) or \(T\) model a Poisson distribution? Justify your answer.
• 15M.2.hl.TZ1.7a: Given that \(2{\text{P}}(X = 4) = {\text{P}}(X = 5)\), show that \(m = 10\).
• 15M.2.hl.TZ1.7b: Given that \(X \le 11\), find the probability that \(X = 6\).
• 15M.2.hl.TZ2.4a: (i)     Find the probability that on a certain day Emma receives more than \(7\) texts. (ii)    ...
• 15M.2.hl.TZ2.4b: Find the probability that Emma receives fewer than \(30\) texts during a week.
• 15N.2.hl.TZ0.11e: The arrival of the elevators at the ground floor between \(08:00\) and \(09:00\) can be modelled...
• 15N.2.hl.TZ0.11f: An elevator can take a maximum of \(10\) workers. Given that \(400\) workers arrive in a half...
• 16M.2.hl.TZ1.10: Students sign up at a desk for an activity during the course of an afternoon. The arrival of each...
• 16M.2.hl.TZ1.13c: Ten balls are taken from the bag. Find the probability that less than four of the balls are...
• 16M.2.hl.TZ1.13d: Find the least number of balls that must be taken from the bag for the probability of taking out...
• 16M.2.hl.TZ1.13e: Another bag also contains balls numbered 1 , 2 or 3. Eight balls are to be taken from this bag...
• 16M.1.hl.TZ2.5a: Find, in terms of \(p\), an expression for \({\text{P}}(X = 4)\).
• 16M.1.hl.TZ2.5b: (i)     Determine the value of \(p\) for which \({\text{P}}(X = 4)\) is a maximum. (ii)     For...
• 16M.2.hl.TZ2.6a: Find the probability that a randomly chosen glass sheet contains at least one flaw.
• 16M.2.hl.TZ2.6b: Find the expected profit, \(P\) dollars, per glass sheet.
• 16M.2.hl.TZ2.6c: Find the probability that it contains no flaws.
• 16N.2.hl.TZ0.3a: Show that...
• 16N.2.hl.TZ0.3b: Given that \({\text{P}}(X = 2) = 0.241667\) and \({\text{P}}(X = 3) = 0.112777\), use part (a) to...
• 17N.1.hl.TZ0.10b.i: Determine the mean of X.
• 17N.1.hl.TZ0.10b.ii: Determine the variance of X.
• 17N.2.hl.TZ0.6a: Find the probability that Lucca eats at least one banana in a particular day.
• 17N.2.hl.TZ0.6b: Find the expected number of weeks in the year in which Lucca eats no bananas.
• 18M.2.hl.TZ1.6: The mean number of squirrels in a certain area is known to be 3.2 squirrels per hectare of...
• 18M.2.hl.TZ2.9a.i: Find the probability that exactly 4 taxis arrive during T.
• 18M.2.hl.TZ2.9a.ii: Find the most likely number of taxis that would arrive during T.
• 18M.2.hl.TZ2.9a.iii: Given that more than 5 taxis arrive during T, find the probability that exactly 7 taxis arrive...
• 18M.2.hl.TZ2.9b: During quiet periods of the day, taxis arrive at a mean rate of 1.3 taxis every 10 minutes. Find...