DP Mathematics SL Questionbank

• Syllabus

5.6

Sub sections and their related questions

Combined events, \(P\left( {A \cup B} \right)\) .

• 12N.2.sl.TZ0.10a: At a large school, students are required to learn at least one language, Spanish or French. It is...
• 12N.2.sl.TZ0.10b: At a large school, students are required to learn at least one language, Spanish or French. It is...
• 12N.2.sl.TZ0.10c(i) and (ii): At a large school, students are required to learn at least one language, Spanish or French. It is...
• 12N.2.sl.TZ0.10d: At a large school, students are required to learn at least one language, Spanish or French. It is...
• 12M.1.sl.TZ2.9a(i) and (ii): Bag A contains three white balls and four red balls. Two balls are chosen at random without...
• 12M.1.sl.TZ2.9b: Bag A contains three white balls and four red balls. Two balls are chosen at random without...
• 12M.1.sl.TZ2.9c: Bag A contains three white balls and four red balls. Two balls are chosen at random without...
• 08N.1.sl.TZ0.5b(i) and (ii): Given that \({\text{P}}(A \cup B) = 0.8\) , (i)     find x ; (ii)    find...
• 08M.1.sl.TZ2.6a(i) and (ii): One student is selected at random. (i)     Calculate the probability that the student is a male...
• 10N.1.sl.TZ0.4a: Write down the value of p .
• 10N.1.sl.TZ0.4b: Find \({\rm{P}}(B)\) .
• 10N.1.sl.TZ0.4c: Find \({\rm{P}}(A'|B)\) .
• 10M.1.sl.TZ1.5a(i), (ii) and (iii): Write down the value of (i)     p ; (ii)    q ; (iii)   r.
• 10M.1.sl.TZ1.5b: Find the value of \({\rm{P}}(A|B')\) .
• 10M.1.sl.TZ1.5c: Hence, or otherwise, show that the events A and B are not independent.
• 10M.1.sl.TZ2.9a(i) and (ii): Find (i)     \({\rm{P}}(E \cap F)\) ; (ii)    \({\rm{P}}(F)\) .
• 10M.1.sl.TZ2.9b(i) and (ii): Find the probability that (i)     José misses his bus and is not late for school; (ii)    José...
• 10M.1.sl.TZ2.9c: The cost for each day that José catches the bus is 3 euros. José goes to school on Monday and...
• 10M.1.sl.TZ2.9d: The cost for each day that José catches the bus is 3 euros. José goes to school on Monday and...
• 09N.2.sl.TZ0.6: Consider the independent events A and B . Given that \({\rm{P}}(B) = 2{\rm{P}}(A)\) , and...
• 10M.2.sl.TZ2.3a: Jan tosses the two dice once. Find the probability that she wins a prize.
• 10M.2.sl.TZ2.3b: Jan tosses the two dice 8 times. Find the probability that she wins 3 prizes.
• 11N.1.sl.TZ0.3a: Write down the probability that the first marble Anna selects is red.
• 11N.1.sl.TZ0.3b: Find the probability that Anna selects two red marbles.
• 11N.1.sl.TZ0.3c: Find the probability that one marble is red and one marble is blue.
• 11M.1.sl.TZ1.8a(i), (ii) and (iii): (i)     Write down the value of s . (ii)    Find the value of q . (iii)   Write down the value...
• 11M.1.sl.TZ1.8b(i) and (ii): (i)     A student is selected at random. Given that the student takes music, write down the...
• 11M.1.sl.TZ1.8c: Two students are selected at random, one after the other. Find the probability that the first...
• 11M.2.sl.TZ1.7a: Find the probability that it passes inspection.
• 11M.2.sl.TZ1.7b: The company would like the probability that a box passes inspection to be 0.87. Find the...
• 08M.1.sl.TZ1.10c: Guiseppi plays a game where he rolls the die. If a blue face lands down, he scores 2 and is...
• 08M.1.sl.TZ1.10e: If the total score is 3, Guiseppi wins \(\$ 10\). If the total score is 2, Guiseppi gets...
• 14M.1.sl.TZ1.5: Celeste wishes to hire a taxicab from a company which has a large number of taxicabs. The...
• 14M.2.sl.TZ2.4b: Find \({\text{P}}(A \cup B)\).
• 14N.1.sl.TZ0.8c: Find the probability that Adam will be late for school.
• 15M.1.sl.TZ1.10c: Find the probability that Ann wins the game.
• 17M.1.sl.TZ1.1a.i: Find the value of \(p\);
• 17M.1.sl.TZ1.1a.ii: Find the value of \(q\).
• 17M.1.sl.TZ1.1b: A girl is selected at random. Find the probability that she takes economics but not history.
• 17M.2.sl.TZ2.10a.i: Find \(q\).
• 17M.2.sl.TZ2.10a.ii: Find \(p\).
• 17M.2.sl.TZ2.10b.i: Write down the probability of drawing three blue marbles.
• 17M.2.sl.TZ2.10b.ii: Explain why the probability of drawing three white marbles is \(\frac{1}{6}\).
• 17M.2.sl.TZ2.10b.iii: The bag contains a total of ten marbles of which \(w\) are white. Find \(w\).
• 17M.2.sl.TZ2.B10c: Jill plays the game nine times. Find the probability that she wins exactly two prizes.
• 17M.2.sl.TZ2.10d: Grant plays the game until he wins two prizes. Find the probability that he wins his second prize...
• 18M.2.sl.TZ1.9a: Find the probability that an orange weighs between 289 g and 310 g.
• 18M.2.sl.TZ1.9b.i: Find the standardized value for 289 g.
• 18M.2.sl.TZ1.9b.ii: Hence, find the value of σ.
• 18M.2.sl.TZ1.9c: To the nearest gram, find the minimum weight of an orange that the grocer will buy.
• 18M.2.sl.TZ1.9d: Find the probability that the grocer buys more than half the oranges in a box selected at random.
• 18M.2.sl.TZ1.9e: The grocer selects two boxes at random. Find the probability that the grocer buys more than half...
• 18M.1.sl.TZ2.8a: Copy and complete the following tree diagram.
• 18M.1.sl.TZ2.8b: Find the probability that Pablo leaves home before 07:00 and is late for work.
• 18M.1.sl.TZ2.8c: Find the probability that Pablo is late for work.
• 18M.1.sl.TZ2.8d: Given that Pablo is late for work, find the probability that he left home before 07:00.
• 18M.1.sl.TZ2.8e: Two days next week Pablo will drive to work. Find the probability that he will be late at least...

Mutually exclusive events: \(P(A \cap B) = 0\) .

• 08N.1.sl.TZ0.5c: Hence, explain why A and B are not mutually exclusive.
• 09N.1.sl.TZ0.8c: Let A be the event that a boy plays football and B be the event that a boy plays rugby. Explain...
• 13N.2.sl.TZ0.4a: Given that \(A\) and \(B\) are mutually exclusive, find \({\text{P}}(B)\).
• 17M.1.sl.TZ1.1a.i: Find the value of \(p\);
• 17M.1.sl.TZ1.1a.ii: Find the value of \(q\).
• 17M.1.sl.TZ1.1b: A girl is selected at random. Find the probability that she takes economics but not history.
• 17M.2.sl.TZ2.10a.i: Find \(q\).
• 17M.2.sl.TZ2.10a.ii: Find \(p\).
• 17M.2.sl.TZ2.10b.i: Write down the probability of drawing three blue marbles.
• 17M.2.sl.TZ2.10b.ii: Explain why the probability of drawing three white marbles is \(\frac{1}{6}\).
• 17M.2.sl.TZ2.10b.iii: The bag contains a total of ten marbles of which \(w\) are white. Find \(w\).
• 17M.2.sl.TZ2.B10c: Jill plays the game nine times. Find the probability that she wins exactly two prizes.
• 17M.2.sl.TZ2.10d: Grant plays the game until he wins two prizes. Find the probability that he wins his second prize...
• 18M.2.sl.TZ1.9a: Find the probability that an orange weighs between 289 g and 310 g.
• 18M.2.sl.TZ1.9b.i: Find the standardized value for 289 g.
• 18M.2.sl.TZ1.9b.ii: Hence, find the value of σ.
• 18M.2.sl.TZ1.9c: To the nearest gram, find the minimum weight of an orange that the grocer will buy.
• 18M.2.sl.TZ1.9d: Find the probability that the grocer buys more than half the oranges in a box selected at random.
• 18M.2.sl.TZ1.9e: The grocer selects two boxes at random. Find the probability that the grocer buys more than half...
• 18M.1.sl.TZ2.8a: Copy and complete the following tree diagram.
• 18M.1.sl.TZ2.8b: Find the probability that Pablo leaves home before 07:00 and is late for work.
• 18M.1.sl.TZ2.8c: Find the probability that Pablo is late for work.
• 18M.1.sl.TZ2.8d: Given that Pablo is late for work, find the probability that he left home before 07:00.
• 18M.1.sl.TZ2.8e: Two days next week Pablo will drive to work. Find the probability that he will be late at least...

Conditional probability; the definition \(P\left( {\left. A \right|B} \right) = \frac{{P\left( {A\mathop \cap \nolimits B} \right)}}{{P\left( B \right)}}\) .

• 12N.2.sl.TZ0.10a: At a large school, students are required to learn at least one language, Spanish or French. It is...
• 12N.2.sl.TZ0.10b: At a large school, students are required to learn at least one language, Spanish or French. It is...
• 12N.2.sl.TZ0.10c(i) and (ii): At a large school, students are required to learn at least one language, Spanish or French. It is...
• 12N.2.sl.TZ0.10d: At a large school, students are required to learn at least one language, Spanish or French. It is...
• 12M.1.sl.TZ2.9a(i) and (ii): Bag A contains three white balls and four red balls. Two balls are chosen at random without...
• 12M.1.sl.TZ2.9b: Bag A contains three white balls and four red balls. Two balls are chosen at random without...
• 12M.1.sl.TZ2.9c: Bag A contains three white balls and four red balls. Two balls are chosen at random without...
• 12M.2.sl.TZ2.7a: A factory makes lamps. The probability that a lamp is defective is 0.05. A random sample of 30...
• 12M.2.sl.TZ2.7b: A factory makes lamps. The probability that a lamp is defective is 0.05. A random sample of 30...
• 08M.1.sl.TZ2.6a(i) and (ii): One student is selected at random. (i)     Calculate the probability that the student is a male...
• 12M.1.sl.TZ1.4a: Write down the value of t .
• 12M.1.sl.TZ1.4b(i) and (ii): (i)     Show that \(r = 0.2\) . (ii)    Write down the value of q and of s .
• 12M.1.sl.TZ1.4c(i) and (ii): (i)     Write down \({\rm{P}}(B')\) . (ii)    Find \({\rm{P}}(A|B')\) .
• 10N.1.sl.TZ0.4a: Write down the value of p .
• 10N.1.sl.TZ0.4b: Find \({\rm{P}}(B)\) .
• 10N.1.sl.TZ0.4c: Find \({\rm{P}}(A'|B)\) .
• 10M.1.sl.TZ1.5a(i), (ii) and (iii): Write down the value of (i)     p ; (ii)    q ; (iii)   r.
• 10M.1.sl.TZ1.5b: Find the value of \({\rm{P}}(A|B')\) .
• 10M.1.sl.TZ1.5c: Hence, or otherwise, show that the events A and B are not independent.
• 10M.1.sl.TZ2.9a(i) and (ii): Find (i)     \({\rm{P}}(E \cap F)\) ; (ii)    \({\rm{P}}(F)\) .
• 10M.1.sl.TZ2.9b(i) and (ii): Find the probability that (i)     José misses his bus and is not late for school; (ii)    José...
• 10M.1.sl.TZ2.9c: The cost for each day that José catches the bus is 3 euros. José goes to school on Monday and...
• 10M.1.sl.TZ2.9d: The cost for each day that José catches the bus is 3 euros. José goes to school on Monday and...
• 09N.1.sl.TZ0.8b: One boy is selected at random. (i)     Find the probability that he plays only one...
• 09M.1.sl.TZ1.2b: Find \({\rm{P}}(B|A)\) .
• 09M.1.sl.TZ1.2c: Find \({\rm{P}}(A \cap B)\) .
• 09M.2.sl.TZ1.7a: Find the probability that the team wins the game.
• 09M.2.sl.TZ1.7b: If the team does not win the game, find the probability that the game was played at home.
• 10M.2.sl.TZ1.10a: Find the probability that a player weighs more than \(82{\text{ kg}}\).
• 10M.2.sl.TZ1.10b(i) and (ii): (i)     Write down the standardized value, z, for \(68{\text{ kg}}\). (ii)    Hence, find the...
• 10M.2.sl.TZ1.10c(i) and (ii): To take part in a tournament, a player’s weight must be within 1.5 standard deviations of the...
• 10M.2.sl.TZ1.10d: Of the players in the league, \(25\% \) are women. Of the women, \(70\% \) take part in the...
• SPNone.1.sl.TZ0.9a(i), (ii) and (iii): (i)     Find \({\rm{P}}(X = 6)\) . (ii)    Find \({\rm{P}}(X > 6)\) . (iii)   Find...
• 13M.1.sl.TZ1.9c: (i)     Write down the probability that the marble is drawn from jar B. (ii)     Given that the...
• 13M.1.sl.TZ1.9d: Given that the marble is red, find the probability that it was drawn from jar A.
• 13M.2.sl.TZ2.9d: Given that at most five of the balls are gold, find the probability that exactly five of the...
• 14M.1.sl.TZ1.9b: Find the probability that Bill wins the first game and Andrea wins the second game.
• 14M.1.sl.TZ1.9d: Given that Bill wins at least one game, find the probability that he wins both games.
• 14M.2.sl.TZ2.10a(ii): A tree is selected at random from the forest. Given that this tree is a giant, find the...
• 13N.2.sl.TZ0.10a: On a randomly selected school day, find the probability that Samantha goes to school by bus.
• 13N.2.sl.TZ0.10b: Given that Samantha went to school by bus on Monday, find the probability that it was raining.
• 14N.1.sl.TZ0.8b: Find the probability that Adam will travel by car and be late for school.
• 14N.1.sl.TZ0.8d: Given that Adam is late for school, find the probability that he travelled by car.
• 14N.2.sl.TZ0.8d: A family is chosen at random. (i)     Find the probability that this family lives in an...
• 14N.2.sl.TZ0.10e: \(25\% \) of the fish in the lake are salmon. \(10\% \) of the salmon are tiddlers. Given that a...
• 15M.2.sl.TZ1.9c: The company changes to the new standard deviation, \(\sigma \), and leaves the mean unchanged. A...
• 15N.2.sl.TZ0.5c: Find \({\text{P}}(C'|D)\).
• 15N.2.sl.TZ0.10c: All the medium and large watermelons are delivered to a grocer. The grocer selects a watermelon...
• 16M.2.sl.TZ1.8a: Find \(k\).
• 16M.2.sl.TZ1.8b: (i)     A day is chosen at random. Write down the probability that Machine A has no...
• 16M.2.sl.TZ1.8c: Find \({\text{E}}(B)\).
• 16M.2.sl.TZ1.8d: (i)     Find the probability that there are exactly two breakdowns on Tuesday. (ii)     Given...
• 17M.2.sl.TZ1.9a: Find \({\text{P}}(X < 9)\).
• 17M.2.sl.TZ1.9b: Find the value of \(\mu \).
• 17N.1.sl.TZ0.1a: Complete the following tree diagram.
• 17N.1.sl.TZ0.1b: Find the probability that exactly one of the selected balls is green.
• 17N.2.sl.TZ0.4a: Find the value of \(k\).
• 17N.2.sl.TZ0.4b: Write down \({\text{P}}(X = 2)\).
• 17N.2.sl.TZ0.4c: Find \({\text{P}}(X = 2|X > 0)\).
• 18M.2.sl.TZ1.5a: Find P(A ∩ B′ ).
• 18M.2.sl.TZ1.5b: Given that P((A ∪ B)′ ) = 0.19, find P(A | B′ ).
• 18M.1.sl.TZ2.8a: Copy and complete the following tree diagram.
• 18M.1.sl.TZ2.8b: Find the probability that Pablo leaves home before 07:00 and is late for work.
• 18M.1.sl.TZ2.8c: Find the probability that Pablo is late for work.
• 18M.1.sl.TZ2.8d: Given that Pablo is late for work, find the probability that he left home before 07:00.
• 18M.1.sl.TZ2.8e: Two days next week Pablo will drive to work. Find the probability that he will be late at least...

Independent events; the definition \(P\left( {\left. A \right|B} \right) = P\left( A \right) = P\left( {\left. A \right|B'} \right)\) .

• 12N.2.sl.TZ0.10a: At a large school, students are required to learn at least one language, Spanish or French. It is...
• 12N.2.sl.TZ0.10b: At a large school, students are required to learn at least one language, Spanish or French. It is...
• 12N.2.sl.TZ0.10c(i) and (ii): At a large school, students are required to learn at least one language, Spanish or French. It is...
• 12N.2.sl.TZ0.10d: At a large school, students are required to learn at least one language, Spanish or French. It is...
• 08N.1.sl.TZ0.5a: Write down an expression for \({\text{P}}(A \cap B)\) .
• 10M.1.sl.TZ1.5a(i), (ii) and (iii): Write down the value of (i)     p ; (ii)    q ; (iii)   r.
• 10M.1.sl.TZ1.5b: Find the value of \({\rm{P}}(A|B')\) .
• 10M.1.sl.TZ1.5c: Hence, or otherwise, show that the events A and B are not independent.
• 09N.1.sl.TZ0.8d: Show that A and B are not independent.
• 09N.2.sl.TZ0.6: Consider the independent events A and B . Given that \({\rm{P}}(B) = 2{\rm{P}}(A)\) , and...
• 11N.2.sl.TZ0.9a: Find the probability that exactly one cap in the sample will be defective.
• 11N.2.sl.TZ0.9b: The sample of caps passes inspection if at most one cap is defective. Find the probability that...
• 11N.2.sl.TZ0.9c(i) and (ii): The heights of the bottles are normally distributed with a mean of \(22{\text{ cm}}\) and a...
• 11N.2.sl.TZ0.9d(i) and (ii): (i)     A bottle is accepted if its height lies between \(21.37{\text{ cm}}\) and...
• 11N.2.sl.TZ0.9e: The bottles and caps are manufactured separately. A sample of 10 bottles and a sample of 10 caps...
• 08M.1.sl.TZ1.10b: If the blue face lands down, the die is not rolled again. If the red face lands down, the die is...
• 08M.1.sl.TZ1.10c: Guiseppi plays a game where he rolls the die. If a blue face lands down, he scores 2 and is...
• 14M.2.sl.TZ2.4a: Find \({\text{P}}(A \cap B)\).
• 14M.2.sl.TZ2.10b: Two trees are selected at random. Find the probability that they are both giants.
• 13N.2.sl.TZ0.4b: Given that \(A\) and \(B\) are independent, find \({\text{P}}(B)\).
• 15N.2.sl.TZ0.5a: Write down an expression for \({\text{P}}(C \cap D)\) in terms of \(k\).
• 16M.2.sl.TZ1.8a: Find \(k\).
• 16M.2.sl.TZ1.8b: (i)     A day is chosen at random. Write down the probability that Machine A has no...
• 16M.2.sl.TZ1.8c: Find \({\text{E}}(B)\).
• 16M.2.sl.TZ1.8d: (i)     Find the probability that there are exactly two breakdowns on Tuesday. (ii)     Given...
• 16M.1.sl.TZ2.8a: (i)     Write down the value of \(p\). (ii)     Find the value of \(q\). (iii)     Write down...
• 16M.1.sl.TZ2.8b: (i)     Write down the probability that this student owns a laptop. (ii)     Find the...
• 16M.1.sl.TZ2.8c: (i)     Copy and complete the following tree diagram. (Do not write on this page.) (ii)    ...
• 16M.2.sl.TZ2.6a: A contestant is chosen at random. Find the probability that their \(S\) score is less than 50.
• 16M.2.sl.TZ2.6b: Given that 1% of the contestants are disqualified, find the value of \(x\).
• 17M.1.sl.TZ1.1a.i: Find the value of \(p\);
• 17M.1.sl.TZ1.1a.ii: Find the value of \(q\).
• 17M.1.sl.TZ1.1b: A girl is selected at random. Find the probability that she takes economics but not history.
• 17N.1.sl.TZ0.1a: Complete the following tree diagram.
• 17N.1.sl.TZ0.1b: Find the probability that exactly one of the selected balls is green.
• 18M.2.sl.TZ1.9a: Find the probability that an orange weighs between 289 g and 310 g.
• 18M.2.sl.TZ1.9b.i: Find the standardized value for 289 g.
• 18M.2.sl.TZ1.9b.ii: Hence, find the value of σ.
• 18M.2.sl.TZ1.9c: To the nearest gram, find the minimum weight of an orange that the grocer will buy.
• 18M.2.sl.TZ1.9d: Find the probability that the grocer buys more than half the oranges in a box selected at random.
• 18M.2.sl.TZ1.9e: The grocer selects two boxes at random. Find the probability that the grocer buys more than half...
• 18M.1.sl.TZ2.8a: Copy and complete the following tree diagram.
• 18M.1.sl.TZ2.8b: Find the probability that Pablo leaves home before 07:00 and is late for work.
• 18M.1.sl.TZ2.8c: Find the probability that Pablo is late for work.
• 18M.1.sl.TZ2.8d: Given that Pablo is late for work, find the probability that he left home before 07:00.
• 18M.1.sl.TZ2.8e: Two days next week Pablo will drive to work. Find the probability that he will be late at least...

Probabilities with and without replacement.

• 08M.1.sl.TZ2.6b: Two students are selected at random. Calculate the probability that neither student plays football.
• 09M.1.sl.TZ1.2b: Find \({\rm{P}}(B|A)\) .
• 09M.1.sl.TZ1.2c: Find \({\rm{P}}(A \cap B)\) .
• 11N.1.sl.TZ0.3a: Write down the probability that the first marble Anna selects is red.
• 11N.1.sl.TZ0.3b: Find the probability that Anna selects two red marbles.
• 11N.1.sl.TZ0.3c: Find the probability that one marble is red and one marble is blue.
• 13M.1.sl.TZ1.9a: Find the probability that   (i)     none of the marbles are green;   (ii)     exactly one...
• 13M.2.sl.TZ2.9a: (i)     Find \({\rm{P}}(X = 0)\) . (ii)     Find \({\rm{P}}(X = 1)\) . (iii)    Hence, find...
• 16M.2.sl.TZ1.8a: Find \(k\).
• 16M.2.sl.TZ1.8b: (i)     A day is chosen at random. Write down the probability that Machine A has no...
• 16M.2.sl.TZ1.8c: Find \({\text{E}}(B)\).
• 16M.2.sl.TZ1.8d: (i)     Find the probability that there are exactly two breakdowns on Tuesday. (ii)     Given...
• 16M.1.sl.TZ2.8a: (i)     Write down the value of \(p\). (ii)     Find the value of \(q\). (iii)     Write down...
• 16M.1.sl.TZ2.8b: (i)     Write down the probability that this student owns a laptop. (ii)     Find the...
• 16M.1.sl.TZ2.8c: (i)     Copy and complete the following tree diagram. (Do not write on this page.) (ii)    ...
• 16M.2.sl.TZ2.6a: A contestant is chosen at random. Find the probability that their \(S\) score is less than 50.
• 16M.2.sl.TZ2.6b: Given that 1% of the contestants are disqualified, find the value of \(x\).
• 17M.1.sl.TZ1.1a.i: Find the value of \(p\);
• 17M.1.sl.TZ1.1a.ii: Find the value of \(q\).
• 17M.1.sl.TZ1.1b: A girl is selected at random. Find the probability that she takes economics but not history.
• 18M.2.sl.TZ1.9a: Find the probability that an orange weighs between 289 g and 310 g.
• 18M.2.sl.TZ1.9b.i: Find the standardized value for 289 g.
• 18M.2.sl.TZ1.9b.ii: Hence, find the value of σ.
• 18M.2.sl.TZ1.9c: To the nearest gram, find the minimum weight of an orange that the grocer will buy.
• 18M.2.sl.TZ1.9d: Find the probability that the grocer buys more than half the oranges in a box selected at random.
• 18M.2.sl.TZ1.9e: The grocer selects two boxes at random. Find the probability that the grocer buys more than half...
• 18M.1.sl.TZ2.8a: Copy and complete the following tree diagram.
• 18M.1.sl.TZ2.8b: Find the probability that Pablo leaves home before 07:00 and is late for work.
• 18M.1.sl.TZ2.8c: Find the probability that Pablo is late for work.
• 18M.1.sl.TZ2.8d: Given that Pablo is late for work, find the probability that he left home before 07:00.
• 18M.1.sl.TZ2.8e: Two days next week Pablo will drive to work. Find the probability that he will be late at least...