3,798 views
2 votes
2 votes
In a lottery, 10 tickets are drawn at random out of 50 tickets numbered from 1 to 50. What is the expected value of the sum of numbers on the drawn tickets?

3 Answers

3 votes
3 votes

we have no's from 1 to 50. So the minimum sum that can be obtained by picking 10 numbers is 1+2+3...10= 55

maximum sum that can be obtained is 41+42+43+.....50=455

so our range of sum is from 55 to 455

now lets assume that no of ways to pick 10 numbers from these 50 numbers is S

We have to calculate the probabilities of the respective sums

so for 55 there is only 1 way of picking 10 numbers such that their sum is 55

similarly there is only 1 way of picking 10 numbers such that the sum is 455

now we have to calculate  where x is the sum and f(x) is the respective probability

now the denominator of this summation will S(as assumed earlier). we only need to worry about the numerator

in the numerator the terms would be like 1*55 + 1*56 +.............+1*454 +1*455. now this sequence is symmetric .So lets pair up the numbers like(1st and last, 2nd and 2nd last ...)  so we will get something like 1*510 +1*510+........ (NOTE : we will have terms like 2*510 ,3*510 etc also)

now if we observe closely this would sum up to 510*(S/2).

so our final answer would be (510*(S/2))/S= 510/2 =255

If this is still not clear how S/2 comes take up smaller examples and see for urself

 

 

edited by
1 votes
1 votes
1 is chosen with probability  $ {49 \choose 9} / {50 \choose 10} = 1/5. $

2 is chosen with probability 1/5.

3 is chosen with probability 1/5. ……..

So the final probability will be $ \sum_{k=1}^{k=50} k / 5= 255$
0 votes
0 votes
expected value of the sum on one number= ((50x51)/2) x (1/50) = 25.5

for 10 numbers =25.5x10=255

Related questions

3 votes
3 votes
1 answer
1
learner_geek asked Jan 20, 2019
638 views
A player tosses two fair coins. He wins rs 2 if 2 heads occur and rs 1 if 1 head occurs. On the other hand, he loses rs 3 if no heads occur. if the player plays 100 times...
0 votes
0 votes
0 answers
2
2 votes
2 votes
1 answer
3
ghostman23111 asked Jan 19, 2019
811 views
Prabha is working in a software company. Her manager is running a dinner for those employees having atleast one son. If Prabha is invited to the dinner and everyone knows...