TIFR CSE 2020 | Part A | Question: 1

Question

Two balls are drawn uniformly at random without replacement from a set of five balls numbered $1,2,3,4,5.$ What is the expected value of the larger number on the balls drawn?

• A. $2.5$
• B. $3$
• C. $3.5$
• D. $4$
• E. None of the above

Answer 1

5 balls we have.

2 balls can be drawn in 10 ways 

Ball1, Ball2

Ball1, Ball3

Ball1, Ball4

Ball1, Ball5

Ball2, Ball3

Ball2, Ball4

Ball2, Ball5

Ball3, Ball4

Ball3, Ball5

Ball4, Ball5

Let x be the larger number on the balls drawn

x	P(x)
1	0
2	1
3	2
4	3
5	4

 

We have to calculate $\sum_{1}^{5} x P(x)$

$1(0) + 2(\frac{1}{10})+ 3(\frac{2}{10})+ 4(\frac{3}{10})+ 5(\frac{4}{10}) = 4$

Answer 2

Total ways of drawing the balls = 5C2 = 10

Note: We consider (x, y) and (y, x) as same, as order doesn't matter here.

Case 1: suppose 1 is present in the balls drawn, then the greater number could be 2, 3, 4, 5

Case 2: suppose 2 is present in the balls drawn, then the greater number could be 3, 4, 5

Case 3: suppose 3 is present in the balls drawn, then the greater number could be 4, 5

Case 3: suppose 4 is present in the balls drawn, then the greater number could be 5 only

Case 5: if 5 is present in one of the balls, this case is already considered in above parts.

Now, expected value of larger number in all the possible draws = (2 + 3 + 4 + 5 + 3 + 4 + 5 + 4 + 5 + 5) / 10 = 40 / 10 = 4

Comments

@Deepak Poonia ,@neeraj_bhatt can you please explain .

If " what is expected value of number on the balls " has been written in the question in place of "what is expected value of a large number on the balls" then still answer would be same ? 

I'm not getting the meaning of asking for larger value on the balls drawn.