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Math Expert V
Joined: 02 Sep 2009
Posts: 58418
Minimum of how many people are needed to have the probability of more  [#permalink]

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Question Stats: 31% (02:00) correct 69% (01:37) wrong based on 302 sessions

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GMAT CLUB TESTS' FRESH QUESTION:

Minimum of how many people are needed to have the probability of more than 1/2 that at lease one of them was born on either on Monday or on Tuesday?

A. 2
B. 3
C. 4
D. 5
E. 6

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Manager  G
Joined: 14 Jun 2018
Posts: 217
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4
1
Suppose there are 2 people A and B
Probability of one on monday/tuesday = 2c1 * 2/7 * 5/7
Prob of both on monday/tuesday = 2/7*2/7
Total = 20/4 + 4/49 = 24/49 < 1/2

Now increase the no of people to 3 : A, B and C
Prob of one on monday/tuesday = 3c1 * 2/7 * 5/7 * 5/7 = 150/343
Prob of two on monday/tuesday = 3c2 * 2/7 * 2/7 * 5/7 = 60/343
Prob of three on monday/tuesday = 2/7* 2/7 * 2/7 = 8/343

Total = 218/343 > 1/2
General Discussion
Math Expert V
Joined: 02 Aug 2009
Posts: 8007
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1
Bunuel wrote:

GMAT CLUB TESTS' FRESH QUESTION:

Minimum of how many people are needed to have the probability of more than 1/2 that at lease one of them was born on either on Monday or on Tuesday?

A. 2
B. 3
C. 4
D. 5
E. 6

Say there are only 2 person..
Prob = $$\frac{2}{7}+\frac{2}{7}-\frac{2}{7*2/7}=\frac{4}{7}-\frac{4}{49}=\frac{24}{49}<\frac{1}{2}$$
Ans will be 3 because with 2, we had answer almost close to 1/2
No calculations required therefore
B
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Manager  B
Joined: 09 Oct 2015
Posts: 225
Re: Minimum of how many people are needed to have the probability of more  [#permalink]

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chetan2u can you please explain a little bit more?

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Manager  S
Joined: 01 Nov 2017
Posts: 95
GMAT 1: 700 Q50 V35 GMAT 2: 640 Q49 V28 GMAT 3: 680 Q47 V36 GMAT 4: 700 Q50 V35 GPA: 3.84
Re: Minimum of how many people are needed to have the probability of more  [#permalink]

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Need Expert to help with this one please!
Intern  B
Joined: 14 Aug 2018
Posts: 2
Re: Minimum of how many people are needed to have the probability of more  [#permalink]

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1
Bunuel wrote:

GMAT CLUB TESTS' FRESH QUESTION:

Minimum of how many people are needed to have the probability of more than 1/2 that at lease one of them was born on either on Monday or on Tuesday?

A. 2
B. 3
C. 4
D. 5
E. 6

My approach was like this:
As, question stated that at least one of the people have to have birthday on either Monday or Tuesday, I need to figure out the probability of all of them having birthday on remaining 5 days of week (other than Monday and Tuesday), then simply asses the value whether it is less than 0.5. If it is less than 0.5 then we have our answer.
First option 2 people: probability of both of them having birthday of remaining 5 days out of 7 days : $$\frac{5}{7} * \frac{5}{7}$$ =$$\frac{25}{49}$$ > 0.5
2nd Option 3 people: probability of all 3 of them having birthday of remaining 5 days out of 7 days : $$\frac{5}{7}*\frac{5}{7}*\frac{5}{7} = \frac{125}{363}$$< 0.5 ---> so 3 people is our answer or option [B]
Manager  G
Status: In last prep stage
Joined: 11 Jun 2017
Posts: 157
GMAT 1: 630 Q44 V33 GMAT 2: 680 Q47 V37 GPA: 3.2
Re: Minimum of how many people are needed to have the probability of more  [#permalink]

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chetan2u wrote:
Bunuel wrote:

GMAT CLUB TESTS' FRESH QUESTION:

Minimum of how many people are needed to have the probability of more than 1/2 that at lease one of them was born on either on Monday or on Tuesday?

A. 2
B. 3
C. 4
D. 5
E. 6

Say there are only 2 person..
Prob = 2/7+2/7-2/7*2/7=4/7-4/49=24/49<1/2
Ans will be 3 because with 2, we had answer almost close to 1/2
No calculations required therefore
B[/quote

Hi Experts,can you please explain Prob = 2/7+2/7-2/7*2/7 .
I am not able to understand this.
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Intern  B
Joined: 06 Nov 2014
Posts: 17
Location: Viet Nam
GMAT 1: 720 Q50 V36 GMAT 2: 740 Q50 V40 GPA: 3.67
Re: Minimum of how many people are needed to have the probability of more  [#permalink]

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3
For "AT LEAST" probability question, it would be a good choice to use P(A) = 1 - P(not A)

Let's say
A - There is at least 1 person born on Monday or Tuesday
B - There is no one born on Monday or Tuesday
Clearly, P(A) = 1 - P(B)

The problem asks when P(A) > 1/2. Or in other words, when P(B) < 1/2.

If we have 1 person, P(B) = 5/7
If we have 2 people, P(B) = (5/7)^2 = 25/49
If we have 3 people, P(B) = (5/7)^3 = something that I am pretty sure that < 1/2

The correct answer is B.3
Math Expert V
Joined: 02 Sep 2009
Posts: 58418
Re: Minimum of how many people are needed to have the probability of more  [#permalink]

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Bunuel wrote:

GMAT CLUB TESTS' FRESH QUESTION:

Minimum of how many people are needed to have the probability of more than 1/2 that at lease one of them was born on either on Monday or on Tuesday?

A. 2
B. 3
C. 4
D. 5
E. 6

_________________
Manager  B
Joined: 27 Nov 2015
Posts: 122
Re: Minimum of how many people are needed to have the probability of more  [#permalink]

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1
chetan2u Bunuel

Kindly guide me where I have gone wrong below.

To be born on either Monday or Tuesday, the probability is 2/7.

So the equation is as follows based on the statment.

x(2/7)>1/2

This leads to x>1.75, hence the number of people needed are a minimum of 2 for the probability to exceed 1/2.
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Location: India
GPA: 3.27
WE: General Management (Retail Banking)
Re: Minimum of how many people are needed to have the probability of more  [#permalink]

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1
Bunuel wrote:

GMAT CLUB TESTS' FRESH QUESTION:

Minimum of how many people are needed to have the probability of more than 1/2 that at lease one of them was born on either on Monday or on Tuesday?

A. 2
B. 3
C. 4
D. 5
E. 6

Assume no of people to be x. The probability of at least one of them is born on Monday or Tuesday is 1- the probability that all of them are born on rest of the days. The probability of one person to be born on the rest of the days=5/7. The probability that all of them are born on the rest of days=(5/7)^x. Required probability= 1-(5/7)^x, which, as per statement is greater than 1/2. So, 1-(5/7)^x=1/2 or (5/7)^x< 1/2, minimum x which satisfies this equation is 3. Hence, 3 or Option B is the answer.
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Manager  B
Joined: 27 Oct 2017
Posts: 72
Re: Minimum of how many people are needed to have the probability of more  [#permalink]

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chetan2u wrote:
Bunuel wrote:

GMAT CLUB TESTS' FRESH QUESTION:

Minimum of how many people are needed to have the probability of more than 1/2 that at lease one of them was born on either on Monday or on Tuesday?

A. 2
B. 3
C. 4
D. 5
E. 6

Say there are only 2 person..
Prob = 2/7+2/7-2/7*2/7=4/7-4/49=24/49<1/2
Ans will be 3 because with 2, we had answer almost close to 1/2
No calculations required therefore
B[/quote

Hi Experts,can you please explain Prob = 2/7+2/7-2/7*2/7 .
I am not able to understand this.

not so sure....but I think we are using the probability OR rule:

prob(A or B) = P(A) + P(B) - P(A and B) ----> 2/7 + 2/7 -(2/7*2/7) ----> 4/7 - 4/49 ---> 24/49 <1/2
Intern  B
Joined: 05 Dec 2017
Posts: 7
GMAT 1: 570 Q48 V20 Minimum of how many people are needed to have the probability of more  [#permalink]

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Assuming the probability that one person was born on a certain day (Mon, Tue, etc) of the week is 1/7.
The probability that one person was born on Monday is equal to the probability that he was born on Tuesday, so the probability that she was born either on Monday or Tuesday is 1/7+1/7=2/7 (let's call it P(A))

Then, the probability that at least one person out of n was born on Monday or Tuesday is (1-(1-P(A))^n). It is the complement of the probability that no one was born on Monday or Tuesday, in other words: 1 - (the probability everyone was born on Wed-Sun): 1 - (5/7)^n

Solving the following equation for n:
1-(1-P(A))^n >= 1/2
1-(5/7)^n >= 1/2
1/2 >= (5/7)^n

You may plug the alternatives and see that if n=2: then 1/2 < 25/49 so try n=3 and you will see that 1/2 >= 125/343. So the answer si you need at least 3 people.
Math Expert V
Joined: 02 Aug 2009
Posts: 8007
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rahulkashyap wrote:
chetan2u can you please explain a little bit more?

Posted from my mobile device

Hi rahulkashyap, nausherwan, AnkitOrYadav

The reason for probability of atleast one of two person, say A and B, to be born on Mon or tuesday is..

So out of 7 days, we are taking just two days - Monday and tuesday..
Probability of A to be born on these 2 days = 2/7
Probability of B to be born on these 2 days = 2/7
But it is possible that both are born in these two days, so we have to subtract it once ..= 2/7 * 2/7 = 4/49
Probability = 2/7 + 2/7 - 4/49
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Re: Minimum of how many people are needed to have the probability of more  [#permalink]

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Minimum of how many people are needed to have the probability of more than 1/2 that at least one of them was born on either on Monday or on Tuesday?

A. 2
B. 3
C. 4
D. 5
E. 6

Minimum of people needed for 100% probability of at least one of them born on either M or T = 6 people (1st born on Wednesday, 2nd born on Thurs.... 6th born on either Mon or Tues)
half of these 6 people = 3 people.

hence at least 3 people are needed to have the probability of more than 1/2.
Choice : B
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Joined: 03 Jun 2019
Posts: 1741
Location: India
Minimum of how many people are needed to have the probability of more  [#permalink]

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Bunuel wrote:

GMAT CLUB TESTS' FRESH QUESTION:

Minimum of how many people are needed to have the probability of more than 1/2 that at lease one of them was born on either on Monday or on Tuesday?

A. 2
B. 3
C. 4
D. 5
E. 6

If n = 1; Probability that at least one of them was born on either on Monday or on Tuesday = 2/7 < 1/2
If n = 2: Probability that at least one of them was born on either on Monday or on Tuesday = 2/7 + 2/7 - (2/7)^2 = 4/7 - 4/49 = 24/49 < 1/2
If n = 3; Probability that at least one of them was born on either on Monday or on Tuesday = 2/7 + 2/7 + 2/7 - 3(5/7)(2/7)^2 + (2/7)^3= 6/7 - 15/343 + 8/343 = 287/343 > 1/2

IMO B
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