MEI Maths Item of the Month
January 2014
An unexpected answer
Mr Teacher sets his class the following problem:
A committee of 3 students is to be chosen from a group of 13 students of which 8 are girls
and 5 are boys. The students are selected at random, without replacement. What is the
expected number of girls on the committee?
Anne Student immediately responds that the answer is
24
.
13
She gives the reason that there are 3 students to be chosen and the proportion of girls is
8
8 24
so she calculated 3  
.
13
13 13
Is she correct?
If proportion of male and female students was different would her method work?
Solution
Yes, she is correct.
8 7 6 28
.
  
13 12 11 143
 8 7 5   8 5 7   5 8 7  210
Similarly P(2)                
.
 13 12 11   13 12 11   13 12 11  429
 8 5 4   5 8 4   5 4 8  120
P(1)                
.
 13 12 11   13 12 11   13 12 11  429
5 4 3
5
.
P(0)    
13 12 11 143
28
210
120 24
Therefore the expected number of girls is 3 
 2
 1

143
429
429 13
The probability of selecting 3 girls is P(3) 
If there were g girls and b boys then the expected number of girls in a committee of 3
would be:
3g ( g  1)( g  2)
6bg ( g  1)
3b(b  1) g


(b  g )(b  g  1)(b  g  2) (b  g )(b  g  1)(b  g  2) (b  g )(b  g  1)(b  g  2)
3g[( g  1)( g  2)  2 b(g  1)  b(b 1)]

(b  g )(b  g  1)(b  g  2)
3g[b 2  2 bg  3b g 2  3g  2]

(b  g )(b 2  2 bg  3b g 2  3g  2)
3g

(b  g )
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v1.1 02/04/14
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