Anonymous
Anonymous asked in Science & MathematicsMathematics · 4 weeks ago

# probability simple problem ?

Five people are in a room. What is the probability that at least two of them were born on the same day of the week?

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• 4 weeks ago

Assuming each of the five people has a different day of birth of the week.

For example the first person can be born on any one of the available seven days; so first person has 7 ways out seven. One day is taken, there six days left; thus second person 6 ways out total seven days of the week…...

1st  person has 7 ways, thus p₁ = 7/7

2nd person has 6 ways, thus p₂ = 6/7

3rd  person has 5 ways, thus p₃ = 5/7

4th  person has 4 ways, thus p₄ = 4/7

5th  person has 3 ways, thus p₅ = 3/7

P( all have different day of the week )

= p₁p₂p₃p₄p₅

= ( 7/7 )( 6/7 )( 5/7 )( 4/7 )( 3/7 )

= 360 / 2401

P( at least two of them were born on the same day of the week )

= 1 - P( all have different day of the week )

= 1 - 360/2401

= 2041 / 2401

—————— ≈ 0.850 or 85.0%

• 4 weeks ago

Let's think of the opposite case. That would mean that they are all born on different days of the week.

The first person has a 7/7 chance of being born on some day of the week. :)

The second person has a 6/7 chance of not being born on the same day as the first person.

The third person has a 5/7 chance of not being born on the same day as either of the first two.

The fourth person has a 4/7 chance...

The fifth person has a 3/7 chance...

So the probability that they are all born on different days of the week is:

7/7 * 6/7 * 5/7 * 4/7 * 3/7

≈ 0.15

But we want the opposite case that at least two people were born on the same day of the week, so subtract from 1.

1 - 0.15

= 0.85

• cosmo
Lv 7
4 weeks ago

Call them A, B, C, D, and E.

A is born on some day of the week.

The probability that B is not born on the same day of the week is 6/7

The probability that C is not born the same day as A or B is 5/7

The probability that D is not born the same day as A, B, or C is 4/7

The probability that E is not born on the same day as A, B, C, or D is 3/7

The probability of all these things being true is (6*5*4*3)/(7^4) = 0.1499

The opposite of that is 1 - 0.1499 = 0.85006

What is the probability if eight people are in the room?