1.	In a group of 16 students, 12 take art and 8 take music. One student takes neither art nor music. The Venn diagram below shows the events art and music. The values p, q, r and s represent numbers of students.

(a)	(i)	Write down the value of s.
(ii)	Find the value of q.
(iii)	Write down the value of p and of r.
(5)
 
(b)	(i)	A student is selected at random. Given that the student takes music, write down the probability the student takes art.
(ii)	Hence, show that taking music and taking art are not independent events.
(4)
 
(c)	Two students are selected at random, one after the other. Find the probability that the first student takes only music and the second student takes only art.
(4)
(Total 13 marks)

2.	Let A and B be independent events such that P(A) = 0.3 and P(B) = 0.8.
(a)	Find P(A ∩ B).
(b)	Find P(A ∪ B).
(c)	Are A and B mutually exclusive? Justify your answer.
(Total 6 marks)

¶ ¬3.	Bag A contains 2 red balls and 3 green balls. Two balls are chosen at random from the bag without replacement. Let X denote the number of red balls chosen. The following table shows the probability distribution for X
X | 0 | 1 | 2
P(X = x) | | |
(a)	Calculate E(X), the mean number of red balls chosen.
(3)
	Bag B contains 4 red balls and 2 green balls. Two balls are chosen at random from bag B.
(b)	(i)	Draw a tree diagram to represent the above information, including the probability of each event.
(ii)	Hence find the probability distribution for Y, where Y is the number of red balls chosen.
(8)
	A standard die with six faces is rolled. If a 1 or 6 is obtained, two balls are chosen from bag A, otherwise two balls are chosen from bag B.
(c)	Calculate the probability that two red balls are chosen.
(5)
(d)	Given that two red balls are obtained, find the conditional probability that a 1 or 6 was rolled on the die.
(3)
(Total 19 marks)

4.	Two boxes contain numbered cards as shown below.
		
	Two cards are drawn at random, one from each box.
 
(a)	Copy and complete the table below to show all nine equally likely outcomes.

3, 9 | |
3, 10 | |
3, 10 | |
(2)
 
	Let S be the sum of the numbers on the two cards.
(b)	Write down all the possible values of S.
(2)
 
(c)	Find the probability of each value of S.
(2)
 
(d)	Find the expected value of S.
(3)
 
(e)	Anna plays a game where she wins $50 if S is even and loses $30 if S is odd.
Anna plays the game 36 times. Find the amount she expects to have at the end of the 36 games.
(3)
(Total 12 marks)

5.	Evan likes to play two games of chance, A and B.
	For game A, the probability that Evan wins is 0.9. He plays game A seven times.
(a)	Find the probability that he wins exactly four games.
(2)
 
	For game B, the probability that Evan wins is p. He plays game B seven times.
(b)	Write down an expression, in terms of p, for the probability that he wins exactly four games.
(2)
(c)	Hence, find the values of p such that the probability that he wins exactly four games is 0.15.
(3)
(Total 7 marks)

· Ø6.	A box holds 240 eggs. The probability that an egg is brown is 0.05.
(a)	Find the expected number of brown eggs in the box.
(2)
 
(b)	Find the probability that there are 15 brown eggs in the box.
(2)
 
(c)	Find the probability that there are at least 10 brown eggs in the box.
(3)
(Total 7 marks)

7.	A company uses two machines, A and B, to make boxes. Machine A makes 60 % of the boxes.
	80 % of the boxes made by machine A pass inspection.
90 % of the boxes made by machine B pass inspection.
	A box is selected at random.

(a)	Find the probability that it passes inspection.
(3)
 
(b)	The company would like the probability that a box passes inspection to be 0.87.
Find the percentage of boxes that should be made by machine B to achieve this.
(4)
(Total 7 marks)

8.	The following table shows the probability distribution of a discrete random variable X.
x | –1 | 0 | 2 | 3
P (X = x) | 0.2 | 10k^2 | 0.4 | 3k
(a)	Find the value of k.
(4)
(b)	Find the expected value of X.
(3)
(Total 7 marks)

9.	A test has five questions. To pass the test, at least three of the questions must be answered correctly.
	The probability that Mark answers a question correctly is . Let X be the number of questions that Mark answers correctly.
 
(a)	(i)	Find E(X).
(ii)	Find the probability that Mark passes the test.
(6)
 
	Bill also takes the test. Let Y be the number of questions that Bill answers correctly.
The following table is the probability distribution for Y.

y | 0 | 1 | 2 | 3 | 4 | 5
P(Y = y) | 0.67 | 0.05 | a + 2b | a – b | 2a + b | 0.04
 
(b)	(i)	Show that 4a + 2b = 0.24.
(ii)	Given that E(Y) = 1, find a and b.
(8)
 
(c)	Find which student is more likely to pass the test.
(3)
(Total 17 marks)

10.	Two fair 4-sided dice, one red and one green, are thrown. For each die, the faces are labelled 1, 2, 3, 4. The score for each die is the number which lands face down.
(a)	List the pairs of scores that give a sum of 6.
(3)
 
	The probability distribution for the sum of the scores on the two dice is shown below.

Sum | 2 | 3 | 4 | 5 | 6 | 7 | 8
Probability | p | q | | | | r |
 
(b)	Find the value of p, of q, and of r.
(3)
 
	Fred plays a game. He throws two fair 4-sided dice four times. He wins a prize if the sum is 5 on three or more throws.
(c)	Find the probability that Fred wins a prize.
(6)
(Total 12 marks)

11.	A van can take either Route A or Route B for a particular journey.
	If Route A is taken, the journey time may be assumed to be normally distributed with mean 46 minutes and a standard deviation 10 minutes.
	If Route B is taken, the journey time may be assumed to be normally distributed with mean µ minutes and standard deviation 12 minutes.
 
(a)	For Route A, find the probability that the journey takes more than 60 minutes.
(2)
 
(b)	For Route B, the probability that the journey takes less than 60 minutes is 0.85.
Find the value of µ.
(3)
 
(c)	The van sets out at 06:00 and needs to arrive before 07:00.
(i)	Which route should it take?
(ii)	Justify your answer.
(3)
 
(d)	On five consecutive days the van sets out at 06:00 and takes Route B. Find the probability that
(i)	it arrives before 07:00 on all five days;
(ii)	it arrives before 07:00 on at least three days.
(5)
(Total 13 marks)

12.	José travels to school on a bus. On any day, the probability that José will miss the bus is .
If he misses his bus, the probability that he will be late for school is .
If he does not miss his bus, the probability that he will be late is .
Let E be the event “he misses his bus” and F the event “he is late for school”.
The information above is shown on the following tree diagram.

(a)	Find
(i)	P(E ∩ F);
(ii)	P(F).
(4)
 
(b)	Find the probability that
(i)	José misses his bus and is not late for school;
(ii)	José missed his bus, given that he is late for school.
(5)

	The cost for each day that José catches the bus is 3 euros. José goes to school on Monday and Tuesday.
(c)	Copy and complete the probability distribution table.

X (cost in euros) | 0 | 3 | 6
P (X) | | |
(3)
 
(d)	Find the expected cost for José for both days.
(2)
(Total 14 marks)

13.	The weights of players in a sports league are normally distributed with a mean of 76.6 kg, (correct to three significant figures). It is known that 80 % of the players have weights between 68 kg and 82 kg. The probability that a player weighs less than 68 kg is 0.05.
(a)	Find the probability that a player weighs more than 82 kg.
(2)
 
(b)	(i)	Write down the standardized value, z, for 68 kg.
(ii)	Hence, find the standard deviation of weights.
(4)
 
	To take part in a tournament, a player’s weight must be within 1.5 standard deviations of the mean.
(c)	(i)	Find the set of all possible weights of players that take part in the tournament.
(ii)	A player is selected at random. Find the probability that the player takes part in the tournament.
(5)
 
	Of the players in the league, 25 % are women. Of the women, 70 % take part in the tournament.
(d)	Given that a player selected at random takes part in the tournament, find the probability that the selected player is a woman.
(4)
(Total 15 marks)

IB Questionbank Maths SL