Homework #12 Due Tuesday, February 14 2017
1a. [5 marks]
Jar A contains three red marbles and five green marbles. Two marbles are drawn from the jar, one after
the other, without replacement.
Find the probability that
(i) none of the marbles are green;
(ii) exactly one marble is green.
Markscheme
(i) attempt to find
eg
,
(M1)
,
A1 N2
(ii) attempt to find
eg
,
(M1)
,
recognizing two ways to get one red, one green (M1)
eg
,
,
A1 N2
[5 marks]
1b. [3 marks]
Find the expected number of green marbles drawn from the jar.
Markscheme
(seen anywhere) (A1)
correct substitution into formula for
A1
1
eg
,
expected number of green marbles is
A1 N2
[3 marks]
1c. [2 marks]
Jar B contains six red marbles and two green marbles. A fair six-sided die is tossed. If the score is or ,
a marble is drawn from jar A. Otherwise, a marble is drawn from jar B.
(i) Write down the probability that the marble is drawn from jar B.
(ii) Given that the marble was drawn from jar B, write down the probability that it is red.
Markscheme
(i)
A1 N1
(ii)
A1 N1
[2 marks]
1d. [6 marks]
Given that the marble is red, find the probability that it was drawn from jar A.
Markscheme
recognizing conditional probability (M1)
eg
,
, tree diagram
attempt to multiply along either branch (may be seen on diagram) (M1)
eg
attempt to multiply along other branch (M1)
eg
adding the probabilities of two mutually exclusive paths (A1)
2
eg
correct substitution
eg
,
A1
A1 N3
[6 marks]
2a. [8 marks]
A bag contains four gold balls and six silver balls.
Two balls are drawn at random from the bag, with replacement. Let
drawn from the bag.
(i) Find
.
(ii) Find
.
(iii) Hence, find
.
Markscheme
METHOD 1
(i) appropriate approach (M1)
eg
,
,
A1 N2
(ii) multiplying one pair of gold and silver probabilities (M1)
eg
,
, 0.24
adding the product of both pairs of gold and silver probabilities (M1)
eg
,
A1 N3
3
be the number of gold balls
(iii)
(seen anywhere) (A1)correct substitution into formula for
,
A1 N3
METHOD 2
(i) evidence of recognizing binomial (may be seen in part (ii)) (M1)
eg
,
correct probability for use in binomial (A1)
eg
,
,
A1 N3
(ii) correct set up (A1)
eg
A1 N2
(iii)
attempt to substitute into
(M1)eg
correct substitution into
A1 N3
[8 marks]
2b. [3 marks]
Hence, find
.
Markscheme
METHOD 1
(seen anywhere) (A1)
correct substitution into formula for
eg
(A1)
,
4
(A1)eg
(A1)eg
A1 N3
METHOD 2
attempt to substitute into
(M1)
eg
correct substitution into
(A1)
eg
A1 N3
[3 marks]
2c. [2 marks]
Fourteen balls are drawn from the bag, with replacement.
Find the probability that exactly five of the balls are gold.
Markscheme
Let
be the number of gold balls drawn from the bag.
evidence of recognizing binomial (seen anywhere) (M1)
eg
,
A1 N2
[2 marks]
2d. [2 marks]
Find the probability that at most five of the balls are gold.
Markscheme
recognize need to find
(M1)
A1 N2
[2 marks]
5
2e. [3 marks]
Given that at most five of the balls are gold, find the probability that exactly five of the balls are gold.
Give the answer correct to two decimal places.
Markscheme
Let
be the number of gold balls drawn from the bag.
recognizing conditional probability (M1)
eg
,
,
,
(A1)
(to dp) A1 N2
[3 marks]
3a. [3 marks]
Bill and Andrea play two games of tennis. The probability that Bill wins the first game is
If Bill wins the first game, the probability that he wins the second game is
.
If Bill loses the first game, the probability that he wins the second game is
.
Copy and complete the following tree diagram. (Do not write on this page.)
6
.
Markscheme
7
A1A1A1 N3
Note: Award A1 for each correct bold probability.
[3 marks]
3b. [2 marks]
Find the probability that Bill wins the first game and Andrea wins the second game.
Markscheme
multiplying along the branches (may be seen on diagram) (M1)
eg
A1 N2
[2 marks]
3c. [4 marks]
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Find the probability that Bill wins at least one game.
Markscheme
METHOD 1
multiplying along the branches (may be seen on diagram) (M1)
eg
adding their probabilities of three mutually exclusive paths (M1)
eg
correct simplification (A1)
eg
A1 N3
METHOD 2
recognizing “Bill wins at least one” is complement of “Andrea wins 2” (R1)
eg finding P (Andrea wins 2)
P (Andrea wins both)
(A1)
evidence of complement (M1)
eg
A1 N3
[4 marks]
3d. [5 marks]
Given that Bill wins at least one game, find the probability that he wins both games.
Markscheme
P (B wins both)
A1
9
evidence of recognizing conditional probability (R1)
eg
correct substitution (A2)
eg
A1 N3
[5 marks]
4a. [2 marks]
Let
and
be independent events, where
Find
and
.
.
Markscheme
correct substitution (A1)
eg
A1 N2
[2 marks]
4b. [2 marks]
Find
.
Markscheme
correct substitution (A1)
eg
A1 N2
[2 marks]
4c. [1 mark]
On the following Venn diagram, shade the region that represents
10
.
Markscheme
A1 N1
4d. [2 marks]
Find
.
Markscheme
appropriate approach (M1)
eg
(may be seen in Venn diagram) A1 N2
[2 marks]
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