Twenty people at a meeting were born during the month of

[AAPL]

Veritas Prep

Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?

A. 10%
B. 33%
C. 67%
D. 90%
E. 99%

OA E

[fskilnik@GMATH]

Veritas Prep

Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?

A. 10%
B. 33%
C. 67%
D. 90%
E. 99%

$$? = 1 - P\left( {\underbrace {{\rm{all}}\,\,20\,\,{\rm{different}}\,\,{\rm{birthday}}\,\,{\rm{dates}}}_{{\rm{unfavorable}}}} \right)$$

$${\rm{Total}}:\,\,30 \cdot 30 \cdot \ldots \cdot 30 = {30^{20}}\,\,\,{\rm{equiprobable}}\,\,{\rm{possibilities}}\,\,\,$$
$${\rm{unfavorable}} = \,\,30 \cdot 29 \cdot \ldots \cdot 11$$
$$P\left( {{\rm{unfavorable}}} \right) = {{30 \cdot 29 \cdot \ldots \cdot 11} \over {{{30}^{20}}}} = 1 \cdot \underbrace {{{29} \over {30}} \cdot {{28} \over {30}} \cdot \ldots {{14} \over {30}}}_{ < < < \,\,1} \cdot \underbrace {{{13} \over {30}} \cdot {{12} \over {30}} \cdot {{11} \over {30}}}_{ \cong \,\,0.05} < < < 0.05 = 5\% $$
$$?\,\,\, > > > \,\,\,100\% - 5\% \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {\rm{E}} \right)$$


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

[Jay@ManhattanReview]

Veritas Prep

Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?

A. 10%
B. 33%
C. 67%
D. 90%
E. 99%

OA E

Probability that at least two people sharing the same birthday = 1 - [Probability that NONE of them sharing the same birthday]

Number of ways of NONE of the 20 shares the birthday = 30P20 = 30!/(30-20)! = 30!/10! = 11*12*......*29*30; we chose permutation since the order of selection matters.

The total number of possible ways of 20 people born in September = 30*30*......*30*30 = 30^20 (Each of the 20 has 30 options);

Probability that at least two people sharing the same birthday = 1 - [(11*12*......*29*30)/(30^20)]

Note that in (11*12*......*29*30)/(30^20), the numerator is significantly less than the denominator; thus, (11*12*......*29*30)/(30^20) is very less.

Or, Probability that at least two people sharing the same birthday = 1 - [(11*12*......*29*30)/(30^20)] = ~0.99.

The correct answer: E

Hope this helps!

-Jay
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[Scott@TargetTestPrep]

Veritas Prep

Twenty people at a meeting were born during the month of September, which has 30 days. The probability that at least two of the people in the room share the same birthday is closest to which of the following?

A. 10%
B. 33%
C. 67%
D. 90%
E. 99%

The probability that at least two of the people in the room share the same birthday is equivalent to subtracting from 1 the probability that no two people in the room share the same birthday.

The first person can have a birthday on any of the 30 days of September. In order to avoid a birthday match, the second person can have a birthday on any of the remaining 29 days. Similarly, to avoid a match with either of the first two people, the third person can have a birthday on any of the remaining 28 days. And so forth, down to the twentieth person. We can then express each event as a probability by dividing by 30, the total number of days in September. The first person's probability of not matching is 30/30 (because they can be born on any day). The second person's probability of not matching the first person is 29/30, and the third person's probability of not matching either of the first two is 28/30. This follows in a similar fashion to the twentieth person.

The probability that no two people in the room share the same birthday (i.e., that they all have different birthdays) is:

30/30 x 29/30 x 28/30 x ... x 11/30
(30 x 29 x 28 x ... x 11)/(30 x 30 x 30 x ... x 30)

30P20 / 30^20 ≈ 0.0002

Therefore, the probability that at least two of the people in the room do share the same birthday is:

1 - 0.0002 = 0.9998 = 99.98%

Answer: E