GATE ECE 2023 | GA | Question: 9

Question

Out of $1000$ individuals in a town, $100$ unidentified individuals are covid positive. Due to lack of adequate covid-testing kits, the health authorities of the town devised a strategy to identify these covid-positive individuals. The strategy is to:

1. Collect saliva samples from all $1000$ individuals and randomly group them into sets of $5$.
2. Mix the samples within each set and test the mixed sample for covid.
3. If the test done in (ii) gives a negative result, then declare all the $5$ individuals to be covid negative.
4. If the test done in (ii) gives a positive result, then all the $5$ individuals are separately tested for covid.

Given this strategy, no more than _______ testing kits will be required to identify all the $100$ covid positive individuals irrespective of how they are grouped.

• A. $700$
• B. $600$
• C. $800$
• D. $1000$

Comments

I think at max we would require $699$ testing kit.

$1st\space 100$ set$\rightarrow 100\space kits$

next $100\space set\rightarrow$ each requires $6$ kits , out of these $100\space set\space 99$ will use $6$ kits but for last set we could use $5$ kits only because till now we have already identified $99$ covid +ve peoples and we know that there are $100$ such people so last set would definitely contain a covid +ve person so we can save $1$ kit by skipping the test mentioned in step $(ii)$. we can directly test each of the $5$ person . So $594+5=599$

hence answer is $\boxed{100+599=699}$

---

@rhl

i. Collect saliva samples from all 1000 individuals and randomly group them into sets of 5.

iv. If the test done in (ii) gives a positive result, then all the 5 individuals are separately tested for covid

Its not upto us to change the size of group, if a person is +ve, all 5 are tested. These are given, and cant be changed.

---

In these types of questions always think of worst case in worst case senerio all the 100 covid positive people will hide themselves in maximum groups that is 100 groups.So out of 200 groups that will be formed at max only 100 groups can be tested out to be positive.and we can run the test on each individual in those group once again that would be 500 more tests because we are testing now each person from that positive groups one by one.

 

pardon my ANGLISH

---

We are given 1000 individuals. Saliva samples are collected from all of them and randomly divided into groups of 5. Therefore, the total number of groups formed is

1000 ÷ 5 = 200 groups.

In the worst-case scenario, assume that 100 individuals are infected and each infected individual belongs to a different group. This means that 100 groups contain one infected person each, while the remaining 200 − 100 = 100 groups contain no infected individuals.

First, all 200 groups are tested. The 100 groups with no infected persons will test negative, requiring 100 testing kits. The remaining 100 groups will test positive, requiring another 100 testing kits. Thus, 200 testing kits are used in the group-testing stage.

Next, each of the 100 groups that tested positive must be tested individually to identify the infected person. Each group contains 5 individuals. In the worst case, the infected individual is detected only after testing all 5 samples in each group. Therefore, the number of testing kits required for individual testing is

100 × 5 = 500 kits.

Finally, adding all the testing kits used,

100 (negative group tests)

• 100 (positive group tests)

• 500 (individual tests)
  = 700 testing kits.

Hence, in the worst-case scenario, a maximum of 700 testing kits will be required.

Answer 1

200 groups made =>200 samples(200 testing kit reqd)
WORST CASE, each group has atmost 1 covid+ve individual
so no of such grps =100
Now, people of these grps are tested individually, so 100*5=500 tests(so 500 testing kit reqd)
so in total, 200+500=700 testing kits are reqd.
opt a

Comments

"so in total, 200+500=700 testing kits are reqd."

@rhl @aravindhz if you understand the above answer, why are we adding 200 here , isn't this considered as overcounting ?

Answer 2

 

Given Data:

• Total individuals = 1000
• Covid-positive individuals = 100
• Individuals are grouped into sets of 5.
• Initial tests are done on 1000 / 5 = 200 mixed samples.

Step-by-step Analysis:

1. First Round of Testing:

   • 200 mixed samples are tested.
   • If a sample tests negative, all 5 members are declared covid-negative.
   • If a sample tests positive, all 5 members are tested individually.

2. Expected Positive Tests in the First Round:

   • There are 100 covid-positive individuals.
   • Each group has 5 people → If a group contains at least one positive individual, the mixed sample will test positive.
   • The number of groups that will test positive depends on how the 100 positive individuals are distributed.
   • Worst Case Scenario: Each group has exactly one covid-positive individual → 100 groups test positive.

3. Second Round of Testing:

   • In these 100 groups, each person is tested separately.
   • Each such group requires 5 individual tests.
   • Total individual tests = 100 × 5 = 500.

Total Test Kits Required:

• First round: 200 tests.
• Second round: 500 tests.
• Total = 200 + 500 = 700 tests.

Answer:

A. 700

Answer 3

Worst case: patient appears in 100 groups out of 200 (Only 100 patients are there to appear so 100 groups)

so first
200 groups will get tested 200*1=200 (multiplied by 1 because we mix all the 5 samples and test only once)

and 100 groups = 100*5 = 500 (each individual needs to be checked. so all 5)

Total = 200+500=700