This one is a data sufficiency question from the topic probability. Tests basic concept of how to find the probability of an event. What is the probability that two students selected to the elocution competition are both boys? Statement 1: The ratio of boys to girls in the class is 3 : 4 Statement 2: There are 11 more girls in the class.

1. GMAT QUANTITATIVE REASONING
ELEMENTARY PROBABILITY
DATA SUFFICIENCY
Diagnostic Test

2. Question
What is the probability that two students selected to the
elocution competition are both boys?
Statement 1: The ratio of boys to girls in the class is 3 : 4
Statement 2: There are 11more girls in the class.

3. Step 1
Get clarity on when the data is sufficient

4. What is the probability that two students selected are both boys?
We will not even look at the statements while answering the following questions
When is the data sufficient and when not?

5. What is the probability that two students selected are both boys?
We will not even look at the statements while answering the following questions
When is the data sufficient and when not?
The data is sufficient if we are able
to get ONE value for the probability.

6. What is the probability that two students selected are both boys?
We will not even look at the statements while answering the following questions
When is the data sufficient and when not?
The data is sufficient if we are able
to get ONE value for the probability.
For instance, if we get more than one
value or if an unknown is part of the
expression, the data is NOT
sufficient.

7. What is the probability that two students selected are both boys?
We will not even look at the statements while answering the following questions
When is the data sufficient and when not? What expression will give the probability ?
The data is sufficient if we are able
to get ONE value for the probability.
For instance, if we get more than one
value or if an unknown is part of the
expression, the data is NOT
sufficient.

8. What is the probability that two students selected are both boys?
We will not even look at the statements while answering the following questions
When is the data sufficient and when not? What expression will give the probability ?
The data is sufficient if we are able
to get ONE value for the probability.
For instance, if we get more than one
value or if an unknown is part of the
expression, the data is NOT
sufficient.
Let the number of boys in the class
be ‘b’ and let there be ‘t’ total
students.

9. What is the probability that two students selected are both boys?
We will not even look at the statements while answering the following questions
When is the data sufficient and when not? What expression will give the probability ?
The data is sufficient if we are able
to get ONE value for the probability.
For instance, if we get more than one
value or if an unknown is part of the
expression, the data is NOT
sufficient.
Let the number of boys in the class
be ‘b’ and let there be ‘t’ total
students.
Probability that two students
selected are both boys =
b(b-1)
t(t-1)

10. Step 2
Let’s evaluate statement 1 alone

11. Statement 1: The ratio of boys to girls in the class is 3 : 4
What is the probability that two students selected are both boys?
· Ratio of boys to girls 3 : 4

12. Statement 1: The ratio of boys to girls in the class is 3 : 4
What is the probability that two students selected are both boys?
· Ratio of boys to girls 3 : 4
If there are 3k boys, there will be 4k
girls and a total of 7k students.
·

13. Statement 1: The ratio of boys to girls in the class is 3 : 4
What is the probability that two students selected are both boys?
· Ratio of boys to girls 3 : 4
If there are 3k boys, there will be 4k
girls and a total of 7k students.
·
We determined in the last slide that
for ‘b’ boys and ‘t’ total students, the
required probability is
b(b−1)
t(t−1)
·

14. Statement 1: The ratio of boys to girls in the class is 3 : 4
What is the probability that two students selected are both boys?
· Ratio of boys to girls 3 : 4
If there are 3k boys, there will be 4k
girls and a total of 7k students.
·
We determined in the last slide that
for ‘b’ boys and ‘t’ total students, the
required probability is
b(b−1)
t(t−1)
·
 the probability =
3k(3k−1)
7k(7k−1)
=
3(3k-1)
7(7k-1)

15. Statement 1: The ratio of boys to girls in the class is 3 : 4
What is the probability that two students selected are both boys?
· Ratio of boys to girls 3 : 4
If there are 3k boys, there will be 4k
girls and a total of 7k students.
·
We determined in the last slide that
for ‘b’ boys and ‘t’ total students, the
required probability is
b(b−1)
t(t−1)
·
 the probability =
3k(3k−1)
7k(7k−1)
=
3(3k-1)
7(7k-1)
Notice that the probability expression comprises a
‘k’ term.
The probability value will depend on the value that
k takes.
So, we CANNOT determine the probability uniquely.

16. Statement 1: The ratio of boys to girls in the class is 3 : 4
Statement 1 alone is NOT sufficient
What is the probability that two students selected are both boys?
· Ratio of boys to girls 3 : 4
If there are 3k boys, there will be 4k
girls and a total of 7k students.
·
We determined in the last slide that
for ‘b’ boys and ‘t’ total students, the
required probability is
b(b−1)
t(t−1)
·
 the probability =
3k(3k−1)
7k(7k−1)
=
3(3k-1)
7(7k-1)
Notice that the probability expression comprises a
‘k’ term.
The probability value will depend on the value that
k takes.
So, we CANNOT determine the probability uniquely.

17. Statement 1: The ratio of boys to girls in the class is 3 : 4
Eliminate choices A and D
Statement 1 alone is NOT sufficient
What is the probability that two students selected are both boys?
· Ratio of boys to girls 3 : 4
If there are 3k boys, there will be 4k
girls and a total of 7k students.
·
We determined in the last slide that
for ‘b’ boys and ‘t’ total students, the
required probability is
b(b−1)
t(t−1)
·
 the probability =
3k(3k−1)
7k(7k−1)
=
3(3k-1)
7(7k-1)
Notice that the probability expression comprises a
‘k’ term.
The probability value will depend on the value that
k takes.
So, we CANNOT determine the probability uniquely.

18. Statement 1: The ratio of boys to girls in the class is 3 : 4
Choices narrow down to B, C or E.
Eliminate choices A and D
Statement 1 alone is NOT sufficient
What is the probability that two students selected are both boys?
· Ratio of boys to girls 3 : 4
If there are 3k boys, there will be 4k
girls and a total of 7k students.
·
We determined in the last slide that
for ‘b’ boys and ‘t’ total students, the
required probability is
b(b−1)
t(t−1)
·
 the probability =
3k(3k−1)
7k(7k−1)
=
3(3k-1)
7(7k-1)
Notice that the probability expression comprises a
‘k’ term.
The probability value will depend on the value that
k takes.
So, we CANNOT determine the probability uniquely.

19. Step 3
Let’s evaluate statement 2 alone.

20. Statement 2 : There are 11more girls in the class.
What is the probability that two students selected are both boys?
There are 11 more girls in the class

21. Statement 2 : There are 11more girls in the class.
What is the probability that two students selected are both boys?
There are 11 more girls in the class
· If the number of boys is 10, there
will be 21 girls and 31 students.

22. Statement 2 : There are 11more girls in the class.
What is the probability that two students selected are both boys?
There are 11 more girls in the class
· If the number of boys is 10, there
will be 21 girls and 31 students.
Probability =
10×9
31×30

23. Statement 2 : There are 11more girls in the class.
What is the probability that two students selected are both boys?
There are 11 more girls in the class
· If the number of boys is 10, there
will be 21 girls and 31 students.
Probability =
10×9
31×30
· If the number of boys is 20, there
will be 31 girls and 51 students.

24. Statement 2 : There are 11more girls in the class.
What is the probability that two students selected are both boys?
There are 11 more girls in the class
· If the number of boys is 10, there
will be 21 girls and 31 students.
Probability =
10×9
31×30
· If the number of boys is 20, there
will be 31 girls and 51 students.
Probability =
20×19
51×50

25. Statement 2 : There are 11more girls in the class.
What is the probability that two students selected are both boys?
There are 11 more girls in the class
· If the number of boys is 10, there
will be 21 girls and 31 students.
Probability =
10×9
31×30
· If the number of boys is 20, there
will be 31 girls and 51 students.
Probability =
20×19
51×50
We are NOT able to
determine the probability
uniquely with this
statement.

26. Statement 2 : There are 11more girls in the class.
What is the probability that two students selected are both boys?
There are 11 more girls in the class
· If the number of boys is 10, there
will be 21 girls and 31 students.
Probability =
10×9
31×30
· If the number of boys is 20, there
will be 31 girls and 51 students.
Probability =
20×19
51×50
We are NOT able to
determine the probability
uniquely with this
statement.
Statement 2 alone is NOT sufficient

27. Statement 2 : There are 11more girls in the class.
Eliminate choice B
What is the probability that two students selected are both boys?
There are 11 more girls in the class
· If the number of boys is 10, there
will be 21 girls and 31 students.
Probability =
10×9
31×30
· If the number of boys is 20, there
will be 31 girls and 51 students.
Probability =
20×19
51×50
We are NOT able to
determine the probability
uniquely with this
statement.
Statement 2 alone is NOT sufficient

28. Statement 2 : There are 11more girls in the class.
Choices narrow down to C or E.
Eliminate choice B
What is the probability that two students selected are both boys?
There are 11 more girls in the class
· If the number of boys is 10, there
will be 21 girls and 31 students.
Probability =
10×9
31×30
· If the number of boys is 20, there
will be 31 girls and 51 students.
Probability =
20×19
51×50
We are NOT able to
determine the probability
uniquely with this
statement.
Statement 2 alone is NOT sufficient

29. Step 4
Let’s combine data from both the
statements.

30. Statements Together : The ratio of boys to girls in the class is 3 : 4 and there are 11 more girls in the class
What is the probability that two students selected are both boys?
We determined that for ‘b’ boys and ‘t’ total
students, the required probability is
b(b−1)
t(t−1)
·

31. Statements Together : The ratio of boys to girls in the class is 3 : 4 and there are 11 more girls in the class
What is the probability that two students selected are both boys?
We determined that for ‘b’ boys and ‘t’ total
students, the required probability is
b(b−1)
t(t−1)
·
From statement 1, we know there are 3k boys
and 4k girls.
·

32. Statements Together : The ratio of boys to girls in the class is 3 : 4 and there are 11 more girls in the class
What is the probability that two students selected are both boys?
We determined that for ‘b’ boys and ‘t’ total
students, the required probability is
b(b−1)
t(t−1)
·
From statement 1, we know there are 3k boys
and 4k girls.
·
From statement 2, we know 4k – 3k = k = 11.·

33. Statements Together : The ratio of boys to girls in the class is 3 : 4 and there are 11 more girls in the class
What is the probability that two students selected are both boys?
We determined that for ‘b’ boys and ‘t’ total
students, the required probability is
b(b−1)
t(t−1)
·
From statement 1, we know there are 3k boys
and 4k girls.
·
So, the class has 33 boys and 44 girls and 77
students.
·
From statement 2, we know 4k – 3k = k = 11.·

34. Statements Together : The ratio of boys to girls in the class is 3 : 4 and there are 11 more girls in the class
What is the probability that two students selected are both boys?
We determined that for ‘b’ boys and ‘t’ total
students, the required probability is
b(b−1)
t(t−1)
·
From statement 1, we know there are 3k boys
and 4k girls.
·
So, the class has 33 boys and 44 girls and 77
students.
·
From statement 2, we know 4k – 3k = k = 11.·
Probability =
33×32
77×76

35. Statements Together : The ratio of boys to girls in the class is 3 : 4 and there are 11 more girls in the class
What is the probability that two students selected are both boys?
We determined that for ‘b’ boys and ‘t’ total
students, the required probability is
b(b−1)
t(t−1)
·
From statement 1, we know there are 3k boys
and 4k girls.
·
So, the class has 33 boys and 44 girls and 77
students.
·
From statement 2, we know 4k – 3k = k = 11.·
Probability =
33×32
77×76
Using the two statements together,
we could determine the probability
uniquely.

36. Statements Together : The ratio of boys to girls in the class is 3 : 4 and there are 11 more girls in the class
What is the probability that two students selected are both boys?
We determined that for ‘b’ boys and ‘t’ total
students, the required probability is
b(b−1)
t(t−1)
·
From statement 1, we know there are 3k boys
and 4k girls.
·
So, the class has 33 boys and 44 girls and 77
students.
·
From statement 2, we know 4k – 3k = k = 11.·
Probability =
33×32
77×76
Using the two statements together,
we could determine the probability
uniquely.
Together the statements are SUFFICIENT.

37. Statements Together : The ratio of boys to girls in the class is 3 : 4 and there are 11 more girls in the class
Eliminate choice E
What is the probability that two students selected are both boys?
We determined that for ‘b’ boys and ‘t’ total
students, the required probability is
b(b−1)
t(t−1)
·
From statement 1, we know there are 3k boys
and 4k girls.
·
So, the class has 33 boys and 44 girls and 77
students.
·
From statement 2, we know 4k – 3k = k = 11.·
Probability =
33×32
77×76
Using the two statements together,
we could determine the probability
uniquely.
Together the statements are SUFFICIENT.

38. Statements Together : The ratio of boys to girls in the class is 3 : 4 and there are 11 more girls in the class
Answer is choice C
Eliminate choice E
What is the probability that two students selected are both boys?
We determined that for ‘b’ boys and ‘t’ total
students, the required probability is
b(b−1)
t(t−1)
·
From statement 1, we know there are 3k boys
and 4k girls.
·
So, the class has 33 boys and 44 girls and 77
students.
·
From statement 2, we know 4k – 3k = k = 11.·
Probability =
33×32
77×76
Using the two statements together,
we could determine the probability
uniquely.
Together the statements are SUFFICIENT.

39. Try this variant
What is the probability that a student selected to the
elocution competition is a boy?
Statement 1: The ratio of boys to girls in the class is 3 : 4
Statement 2: There are 11more girls in the class.

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