How many randomly assembled people are needed to have a better than 50

The answer is C:3.

Essentially, you need to find the number of people so that the probability of all of them not being from a leap year is LESS THAN 50%. The probability of any one person not being born on a leap year is approximately 3/4.

So you are essentially solving for:

\((\frac{3}{4})^n < \frac{1}{2}\)

n = 1: 3/4 < 1/2? NO
n = 2: (3/4)(3/4) = 9 / 16 < 1/2? NO
n = 3 (3/4)(3/4)(3/4) = 27 / 64 < 1/2? YES

Therefore the answer is 3.

My solution

Prob(at least one of them was born in a leap year) > 50%
P(A) = 1 - P(A')
1 - Prob(none were born in a leap year) > 1/2
Prob(none) - 1 < -1/2
Prob(none) < 1/2

Assume that there are n people.
Prob(none) = (3/4)^n
(3/4)^n < 1/2
3^n < (4^n)/2
3^n < (2^2n)/2
3^n < 2^(2n-1)

If n = 3, 3^3 < 2^5 is true (27 < 32)

Thus, The answer is 3

This is how I would ans .

P(people born in leap year) = 1/4 (because out of every 4 years 1 is a leap year )
P(People not born in leap year ) = 3/4

Now for any n no. of people (3/4)^n is Probability of those people who were not born in leap year

As total probability is 1

1-(3/4)^n represents the cumulative probability of all those people born in leap year ( Eg this P includes probability of 1 person in group to be
born in leap year to n persons in a group to be born in leap year - which ans at least 1 person is born in leap year )

So 1-(3/4)^n>1/2

after trying diff values of n , it comes out as 3

Bunuel
how is Probability of a randomly selected person NOT to be born in a leap year is \(\frac{3}{4}\). ???

if you consider years 1896 1897 1898 1899 1900 1901 1902 1903 1904 .... Here 1896 and 1904 are the only leap years.
How did you generalize probability not being a leap year to be 3/4 ?
And also question didn't ask about minimum number of people needed. Ans can be 3 or 4 or 5

Yes, every 100's year is not leap but for this question it's implied that 1 in 4 years is leap.

If you have 1 person, the chance that he's born in a leap year is \(\frac{1}{4}\)
If you have 2 persons the chances are \(\frac{2}{4}\) = 50%
We need AT LEAST 50%, so 1 extra person will do the job.

We need 3 persons.

C.

Keeping it simple:

P (A person was born in leap year) = 1/4
This is less than 50% - not the answer

P (A or B was born in a leap year) = P (A born in leap year) + P (B born in leap year) - P (Both born in leap year)


The two events are independent. Probability that A was born is a leap year is independent of the probability that B was born in a leap year.
So P(Both) = P(A)*P(B)

P (A or B was born in a leap year) = 1/4 + 1/4 - (1/4)*(1/4) = 1/2 - 1/16

This is less than 50% but close - not the answer

So when you pick, three people, the probability of someone born in a leap year will be higher than 50%.

If you want to calculate it, you can do it as given below:

P(A or B or C was born in a leap year) = 1/4 + 1/4 + 1/4 - 1/16 - 1/16 - 1/16 + 1/64 = 1/2 + 1/16 + 1/64 (same as the sets concept)

This is definitely more than 50%.

Answer (C)

Check out this video: https://youtu.be/0BCqnD2r-kY

Or we can reverse the problem. If the probability someone is born in a leap year is greater than 1/2, then the probability no one was born in a leap year needs to be less than 1/2.

If we pick one person, the probability they weren't born in a leap year is 3/4. If we pick two people, the probability both were not born in a leap year is (3/4)(3/4) = (3/4)^2 = 9/16. That's still bigger than 1/2, but if we pick a third person, the probability none of the three people was born in a leap year is (3/4)^3 which is less than 1/2, which means the probability someone was born in a leap year is now greater than 1/2. So the answer is three.

Every 4 consecutive years have 1 Leap year and 3 normal years

i.e. Probability of an year to be leap year = 1/4
and Probability of an year to NOT to be leap year = 1/4

Probability of 1 randomly selected year to be NON leap year = (3/4) i.e. Greater than 50%

Probability of 2 randomly selected year to be NON leap year = (3/4)*(3/4) = 9/16 i.e. Greater than 50%

Probability of 3 randomly selected year to be NON leap year = (3/4)*(3/4)*(3/4) = 27/64 i.e. Less than 50% i.e. there are more than 50% chances that atleast one of the selected 3 years will be a Leap Year

Hence - 3 Years

Answer: option C

#If one is selected P(A born in a leap year) = 1/4 or 25%

#If two are selected the probability that at least one is born in a leap year P(A only, B only, or A & B) = P(A) + P(B) + P(A&B) = (1/4 * 3/4) + (3/4 * 1/4) + (1/4 * 1/4) = 7/16

From here I just made the assumption that 3 people would push it above 50% and selected C.

I am just curious of the method that VeritasPrepKarishma used... Maybe there is a Thread or Blog post you could point me to that explains this in more detail.

Thanks in advance!

This method is a counterpart of SETS formulas we use.

10 people like A and 20 people like B. 5 like both A and B. So how many people like A or B?

n(A or B) = n(A) + n(B) - n(A and B)
We subtract n(A and B) because it is counted twice - once in n(A) and another time in n(B). But we want to count it only once so subtract it out once.
n(a or B) gives us the number of people who like at least one of A and B.

Similarly, we can use this method for probability
p(A or B) = p(A) + p(B) - p(A and B)

It is exactly the same concept. It gives us the probability that at least one of two people are born in a leap year.
When considering p(A), p(B) is to be ignored and when c considering p(B), p(A) is to be ignored. We take care of both when we subtract p(A and B).

Similarly, we can use the sets concept for 3 people using the three overlapping sets formula:

n(A or B or C) = n(A) + n(B) + n(C) - n(A and B) - n(B and C) - n(C and A) + n(A and B and C)

it becomes
p(A or B or C) = p(A) + p(B) + p(C) - p(A and B) - p(B and C) - p(C and A) + p(A and B and C)

Also, check out this video: https://youtu.be/0BCqnD2r-kY

good question. But i doubt whether we can solve like this

#If one is selected P(A born in a leap year) = 1/4 or 25%
#If two is selected P(A,B born in a leap year) = 2/4

Since we need more than 50% we pickup 3 as it is minimal among others.

Hi Mechmeera,

Highlighted step is the wrong step that you have written

Because when you select two individuals then one of them can be born in Leap Year and other is not and vise versa as well

So Probability of both A and B born in Leap year = (1/4)for A * (1/4)for B = 1/16

But Probability of One of them Born in leap year = (1/4)*(3/4) + (1/4)*(3/4) = 6/16

i.e. Probability that atleast one of A and B born in Leap year = (1/16)+(6/16) = 7/16 which is LESS than 50%

But when you follow the same process for 3 people then the probability exceeds 50%

I hope it helps!

The thought process is correct but the execution is not.
The method I have used is the same as this. When two are selected, the probability that at least one of them has bday in a leap year is 2/4 - 1/16, not only 2/4. I have explained 'why' here: what-is-the-minimum-number-of-randomly-chosen-people-needed-in-order-t-203018.html#p1559070

Hi Bunuel,

What is wrong in the following approach?

Cases considered :
1. Born in a leap year
2. Not born in a leap year

In that case, the probability of not born in a leap year =1/2

And subsequently, the probability that at least one of them is born in a leap year would turn out to be greater than 50% for 2 people.

Please help.

Thank you.

Let me ask you: is the probability that you win a lottery 1/2?
1. You win the lottery
2. You won't win the lottery

Or: what is the probability to meet a dinosaur on the street? Is it 1/2? Either you meet it or not?

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