HSC Worked Solutions
09
(i)
projectmaths.com.au
4a
A test consists of five multiple-choice questions. Each question has four
alternative answers. For each question only one of the alternative answers is
correct. Huong randomly selects an answer to each of the five questions.
(i)
What is the probability that Huong selects three correct and two
2
incorrect answers?
(ii)
What is the probability that Huong selects three or more correct
2
answers?
(iii)
What is the probability that Huong selects at least one incorrect
1
answer?
(ii)
P(3 or more correct)
3
Let p = P(incorrect) =
= P(3 correct) + P(4 correct)
4
+ P(5 correct)
1
Let q = P(correct) =
 5   3 2  1 3
5   3   1 4
4
=      +    
3  4   4 
 4  4   4 
As (p + q)5 = ... + 5C p2q3 + ...
 
 
3
 5   1 5
+   
5  4 
 
45
15
1
=
+
+
512
1024
1024
53
=
512
P(3 correct, 2 incorrect) = 5C 3 p q
2 3
5
=  
3
 
= 10 ×
=
2
3 1
   
4 4
45
512
3
1
9
×
64
16
(iii) P(at least one incorrect) = 1 – P(all correct)
5
1
=1–  
4
1
=1–
1024
1023
=
1024
* These solutions have been provided by projectmaths and are not supplied or endorsed by the Board of Studies
Board of Studies: Notes from the Marking Centre
Source: http://www.boardofstudies.nsw.edu.au/hsc_exams/
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