Eight people have to go to the hospital during a particular week. What

Question

Eight people have to go to the hospital during a particular week. What is the probability that on at least one of the days, at least two people will go to the hospital?

A. 0.2
B. 0.3
C. 0.7
D. 0.8
E. 1

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jack5397 · Answer

IMO - E

As there are 8 people and 7 days in a week one day 2 or more than 2 people have to go to the hospital in order to satisfy the condition.

DrAnkita91 · Answer

please can you tell me how did you get the answer of it?

Bunuel · Answer

Eight people have to go to the hospital during a particular week. What is the probability that on at least one of the days, at least two people will go to the hospital?

A. 0.2
B. 0.3
C. 0.7
D. 0.8
E. 1

There are 7 days and 8 people. Thus, it's not possible to distribute them in such a way that on  each  day less than 2 people go to the hospital. Therefore, the probability that on at least one of the days, at least two people will go to the hospital is 1.

Answer: A.

luisdicampo · Answer

Deconstructing the Question

We have  8  people and only  7  days in the week.

We want the probability that on at least one day, at least  2  people go to the hospital.

The fastest way is to think about the complement: no two people go on the same day. But that would require all  8  people to go on different days.

That is impossible, because there are only  7  days.

Step-by-step

To avoid having at least  2  people on the same day, each of the  8  people would need a different day.

But there are only  7  days available.

So once  7  people are assigned to different days, the  8th  person must share a day with someone else.

Therefore, at least one day must have at least  2  people.

So the probability is:

1

Answer: E) 1

vasu1104 · Answer

first time encountering this type of que. i was not able to solve myself but learnt form AI chat bot. so trying to share what i have learnt and how to approach it.

so given that total people = 8
total num of days =7

now its asking for probability that on at least one of the day at least two people will go to hospital.
if try to find this directly, it would be too time consuming and still wont get it.

so better to go opposite side that is at most one person on each day.
now since we have 8 people and 7 days. so we can have only 7 people going to hospital. so it means its not possible to have all 8 people one each since one person will left out and to accommodate that person, we would need to have 2 people on one day. nmow this is not possible as this is not what we are looking. so finding this probability is 0.

p(happens) = 1- P (doesnt happen)
1- 0 = 1

option E

onlyPlanA · Answer

Intuitively, I understood why the answer is 1. But mathematically, I still can't figure out. How would have we approached this question, if there were 15 people and exactly three of them will go to the hospital on a certain day?

egmat · Answer

Hi onlyPlanA,

Your intuition  is  the math here. There's no probability formula hiding behind this one. The rigorous version is just a counting-capacity argument (the pigeonhole idea), and once you see it as a formula you can run it on any variant.

The general approach

The event is "at least one day has at least  k  people." Instead of computing it, ask the opposite:  how many people can I fit while never letting any day reach k?

- Each day is allowed at most  (k - 1)  people before it trips the event.
- With  D  days, the maximum you can pack without ever hitting k is:

capacity = D x (k - 1)

- If the  actual number of people > capacity , it's  impossible  to avoid the event, so the probability is  1 .

Your original question

Here k =  2  (at least two on a day), D =  7  days.

- Capacity =  7  x ( 2  -  1 ) =  7  people.
- You have  8  people, and  8  >  7  - can't avoid it - probability =  1 . That's choice  E .

Your variant - 15 people, threshold of 3

Now k =  3  (at least three on one day), still D =  7  days.

- Capacity =  7  x ( 3  -  1 ) =  14  people.
- You have  15  people, and  15  >  14  - again impossible to spread them out - probability =  1 .

Same machine, different numbers.

The takeaway

When a probability question gives you more items than "slots times the allowed-per-slot," the answer is forced to  1  - you never actually compute a fraction. The moment the people count slips  under  capacity (say  14  people, threshold  3 ), it stops being certain and you'd genuinely have to count arrangements. Spotting which side of that line you're on is the whole skill.

Answer: E

Eight people have to go to the hospital during a particular week. What

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Eight people have to go to the hospital during a particular week. What

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