Out of 13 applicants for a job, there are 5 women and 8 men.

Question

Out of 13 applicants for a job, there are 5 women and 8 men. Two persons are to be selected for the job. The probability that at least one of the selected persons will be a woman is:

A.16/39
B.34/39
C.25/39
D.16/53
E.27/53

gmatophobia · Answer

Probability(at least one women) = 1 - Probability(both men)

Probability of both men =  \frac{^8C_2}{^{13}C_2} = \frac{14}{39}

Probability of at least one women =  1 - \frac{14}{39} = \frac{25}{39}

Option C

Crater · Answer

At least one woman =\frac{(^5C_2+^5C_1*^8C_1)}{^{13}C_2}=\frac{(10+40)}{75}=\frac{25}{39}

Option (C)

GSV · Answer

There are 2 cases - 1 women , 1men or 2 women 
(5C1*8C1 + 5C2 )/ 13C2 = ((5*4 )+ 5 )/ (13*3) = 25/39

egmat · Answer

This is an  "at least one" probability  problem, and the fastest way to solve these is using the  complement trick :

P(at least one woman) =  1  - P(NO women)

P(NO women) means BOTH selected persons are men.

Step 1:  Find total ways to choose  2  people from  13 .
 C( 13 , 2 )  = ( 13  ×  12 ) / ( 2  ×  1 ) =  78

Step 2:  Find ways to choose  2  men from  8  men (this is the "no women" scenario).
 C( 8 , 2 )  = ( 8  ×  7 ) / ( 2  ×  1 ) =  28

Step 3:  P(no women) =  28 / 78  =  14 / 39

Step 4:  P(at least one woman) =  1  -  14 / 39  =  25 / 39

Answer: C

Why use the complement? "At least one woman" includes  2  separate cases — exactly  1  woman OR exactly  2  women. Calculating both and adding them works, but it's slower. The complement ("no women at all") is just one simple case, so subtracting it from  1  is much faster.

General principle:  Whenever you see "at least one" in a probability question, immediately think  complement . Calculate the probability of the opposite event (none), and subtract from  1 . This shortcut saves significant time on the GMAT.

The most common mistake  here is forgetting to use  combinations  and instead just multiplying individual probabilities incorrectly. Always count the combinations carefully: total outcomes in the denominator, favorable outcomes in the numerator.

Out of 13 applicants for a job, there are 5 women and 8 men.

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Out of 13 applicants for a job, there are 5 women and 8 men.

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