Date | May 2018 | Marks available | 2 | Reference code | 18M.2.SL.TZ1.S_2 |
Level | Standard Level | Paper | Paper 2 | Time zone | Time zone 1 |
Command term | Question number | S_2 | Adapted from | N/A |
Question
A biased four-sided die is rolled. The following table gives the probability of each score.
Find the value of k.
Calculate the expected value of the score.
The die is rolled 80 times. On how many rolls would you expect to obtain a three?
Markscheme
* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.
evidence of summing to 1 (M1)
eg 0.28 + k + 1.5 + 0.3 = 1, 0.73 + k = 1
k = 0.27 A1 N2
[2 marks]
correct substitution into formula for E (X) (A1)
eg 1 × 0.28 + 2 × k + 3 × 0.15 + 4 × 0.3
E (X) = 2.47 (exact) A1 N2
[2 marks]
valid approach (M1)
eg np, 80 × 0.15
12 A1 N2
[2 marks]
Examiners report
Syllabus sections
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22M.1.SL.TZ2.9d:
Determine the value of bb.
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22M.1.SL.TZ2.9e:
Find the value of aa, providing evidence for your answer.
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22M.2.SL.TZ1.4a:
Show that 2k2-k+0.12=02k2−k+0.12=0.
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17M.2.SL.TZ2.S_10b.i:
Write down the probability of drawing three blue marbles.
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18M.2.SL.TZ1.S_2b:
Calculate the expected value of the score.
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18M.2.SL.TZ2.S_10d.i:
Find the expected number of bags in this crate that contain at most one small apple.
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19M.2.AHL.TZ2.H_10b:
Copy and complete the probability distribution table for Y.
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17M.2.SL.TZ2.S_10b.iii:
The bag contains a total of ten marbles of which ww are white. Find ww.
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19M.2.SL.TZ1.S_10c.i:
Write down the value of xx.
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18M.2.SL.TZ2.S_10d.ii:
Find the probability that at least 48 bags in this crate contain at most one small apple.
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17M.2.SL.TZ2.S_10d:
Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.
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18N.1.SL.TZ0.S_9a.ii:
Find the probability, in terms of nn, that the game will end on her second draw.
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18M.1.AHL.TZ2.H_3a:
Find the value of p.
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17N.2.SL.TZ0.S_4a:
Find the value of kk.
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19M.2.SL.TZ1.S_10c.ii:
Hence, find the value of yy.
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18N.1.SL.TZ0.S_9b.ii:
fourth draw.
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19N.2.AHL.TZ0.H_10b:
Sketch the graph of ff. State the coordinates of the end points and any local maximum or minimum points, giving your answers in terms of aa.
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19N.2.AHL.TZ0.H_10c.iii:
the median of XX.
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17M.1.SL.TZ1.S_10b:
Given that tanθ>0tanθ>0, find tanθtanθ.
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18M.1.AHL.TZ1.H_3b:
Find the expected value of T.
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18M.2.SL.TZ2.S_10c:
Find the probability that a bag of apples selected at random contains at most one small apple.
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18M.2.SL.TZ2.S_10a.ii:
Show that μ = 106.
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17N.2.SL.TZ0.S_4b:
Write down P(X=2)P(X=2).
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18N.1.SL.TZ0.S_9b.i:
third draw.
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21N.2.SL.TZ0.9c.iii:
premium.
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18M.2.SL.TZ2.S_10a.i:
Write down the value of k.
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17M.2.SL.TZ2.S_10a.i:
Find qq.
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19N.2.SL.TZ0.S_7b:
Find the difference between the greatest possible expected value and the least possible expected value.
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16N.2.AHL.TZ0.H_11b:
Find the values of the constants aa and bb.
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17M.1.SL.TZ1.S_10c:
Let y=1cosxy=1cosx, for 0<x<π20<x<π2. The graph of yybetween x=θx=θ and x=π4x=π4 is rotated 360° about the xx-axis. Find the volume of the solid formed.
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16N.2.AHL.TZ0.H_1a:
Determine the value of E(X2)E(X2).
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18N.1.SL.TZ0.S_9a.i:
Find the probability, in terms of nn, that the game will end on her first draw.
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19M.2.AHL.TZ2.H_10d:
In any given year of 365 days, the probability that Steffi does not visit Will for at most nn days in total is 0.5 (to one decimal place). Find the value of nn.
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19N.2.AHL.TZ0.H_10c.ii:
E(X)E(X).
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EXN.1.SL.TZ0.3:
The following table shows the probability distribution of a discrete random variable XX where x=1, 2, 3, 4x=1,2,3,4.
Find the value of kk, justifying your answer.
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22M.1.SL.TZ2.9b:
Find P(X>2)P(X>2).
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16N.2.AHL.TZ0.H_11e:
Determine the minimum value of xx such that the probability Kati receives at least one free gift is greater than 0.5.
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18M.1.AHL.TZ1.H_3a:
Find the value of a and the value of b.
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19N.2.AHL.TZ0.H_10a:
Find, in terms of aa, the probability that XX lies between 1 and 3.
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16N.2.SL.TZ0.S_7b:
Let XX be the number of red discs selected. Find the smallest value of mm for which Var(X )<0.6Var(X )<0.6.
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17M.2.SL.TZ2.S_10c:
Jill plays the game nine times. Find the probability that she wins exactly two prizes.
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21M.1.SL.TZ1.9a:
Find the value of pp.
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21M.1.SL.TZ1.9b:
Hence, find the value of E(X) E(X) .
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21M.1.SL.TZ1.9c.i:
State the range of possible values of rr.
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21M.1.SL.TZ1.9d:
Hence, find the range of possible values for E(Y)E(Y).
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21M.2.SL.TZ2.9a:
Find the value of cc.
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21M.2.SL.TZ2.9c:
Given that the grand prize is not won and the grand prize continues to double, write an expression in terms of nn for the value of the grand prize in the nthnth week of the lottery.
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16N.2.SL.TZ0.S_7a:
Write down the probability that the first disc selected is red.
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18M.1.AHL.TZ2.H_3b.i:
Find μ, the expected value of X.
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18M.1.AHL.TZ2.H_3b.ii:
Find P(X > μ).
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19M.1.SL.TZ2.S_1b:
Find E(X)E(X).
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16N.2.AHL.TZ0.H_11a:
Show that P(X=3)=0.001P(X=3)=0.001 and P(X=4)=0.0027P(X=4)=0.0027.
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19M.2.SL.TZ1.S_10a.ii:
Find the probability of rolling two or more red faces.
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19M.2.AHL.TZ2.H_10c:
Hence find the expected number of times per day that Steffi is fed at Will’s house.
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16N.2.AHL.TZ0.H_11c:
Deduce that P(X=n)P(X=n−1)=0.9(n−1)n−3P(X=n)P(X=n−1)=0.9(n−1)n−3 for n>3n>3.
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16N.1.AHL.TZ0.H_2a:
Complete the probability distribution table for XX.
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16N.2.AHL.TZ0.H_1b:
Find the value of Var(X)Var(X).
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19M.1.SL.TZ2.S_1a:
Find the value of kk.
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18N.1.SL.TZ0.S_9c:
Hayley plays the game when nn = 5. She pays $20 to play and can earn money back depending on the number of draws it takes to obtain a blue marble. She earns no money back if she obtains a blue marble on her first draw. Let M be the amount of money that she earns back playing the game. This information is shown in the following table.
Find the value of kk so that this is a fair game.
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19N.2.AHL.TZ0.H_10c.i:
aa.
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17M.2.SL.TZ2.S_10b.ii:
Explain why the probability of drawing three white marbles is 1616.
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21M.2.SL.TZ2.9d:
The wthwth week is the first week in which the player is expected to make a profit. Ryan knows that if he buys a lottery ticket in the wthwth week, his expected profit is $p$p.
Find the value of pp.
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21M.1.SL.TZ1.9c.ii:
Hence, find the range of possible values of qq.
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19M.2.SL.TZ1.S_10b:
Show that, after a turn, the probability that Ted adds exactly $10 to his winnings is 1313.
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19N.1.AHL.TZ0.H_1b:
Given that E(X)=10E(X)=10, find the value of NN.
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20N.1.AHL.TZ0.H_1:
A discrete random variable XX has the probability distribution given by the following table.
Given that E(X)=1912E(X)=1912, determine the value of pp and the value of qq.
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21N.2.SL.TZ0.9c.ii:
large.
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21N.2.SL.TZ0.9a:
Find P(μ-1.5σ<X<μ+1.5σ)P(μ−1.5σ<X<μ+1.5σ).
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21N.2.SL.TZ0.9b:
Find the value of μ and of σ.
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21N.2.SL.TZ0.9d:
The selling prices of the different categories of avocado at this supermarket are shown in the following table:
The supermarket pays the farm $ 200 for the avocados and assumes it will then sell them in exactly the same proportion as purchased from the farm.
According to this model, find the minimum number of avocados that must be sold so that the net profit for the supermarket is at least $ 438.
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22M.1.SL.TZ2.9a:
Show that p=0.4 and q=0.2.
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17M.2.SL.TZ2.S_10a.ii:
Find p.
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19N.2.SL.TZ0.S_7a:
Show that b=0.3−a.
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19M.2.SL.TZ1.S_10a.i:
Find the probability of rolling exactly one red face.
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16N.1.AHL.TZ0.H_2b:
Find the expected value of X.
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17M.1.SL.TZ1.S_10a:
Show that cosθ=34.
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20N.2.SL.TZ0.S_3b.i:
Find the value of p which gives the largest value of E(X).
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20N.2.SL.TZ0.S_3b.ii:
Hence, find the largest value of E(X).
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18M.2.SL.TZ1.S_2a:
Find the value of k.
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19M.2.SL.TZ1.S_10d:
Ted will always have another turn if he expects an increase to his winnings.
Find the least value of w for which Ted should end the game instead of having another turn.
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19N.1.AHL.TZ0.H_1a:
Find the value of p.
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20N.2.SL.TZ0.S_3a:
Find an expression for q in terms of p.
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19M.2.AHL.TZ2.H_10e:
Show that the expected number of occasions per year on which Steffi visits Will’s house and is not fed is at least 30.
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16N.2.AHL.TZ0.H_11d:
(i) Hence show that X has two modes m1 and m2.
(ii) State the values of m1 and m2.
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18M.2.SL.TZ2.S_10b:
Find P(M < 95) .
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17N.2.SL.TZ0.S_4c:
Find P(X=2|X>0).
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19M.2.AHL.TZ2.H_10a:
Find the probability that on a randomly selected day, Steffi does not visit Will’s house.
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21M.1.SL.TZ1.9e:
Agnes and Barbara play a game using these dice. Agnes rolls die A once and Barbara rolls die B once. The probability that Agnes’ score is less than Barbara’s score is 12.
Find the value of E(Y).
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21M.2.SL.TZ2.9b:
Determine whether this lottery is a fair game in the first week. Justify your answer.
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21N.2.SL.TZ0.9c.i:
medium.