A shipment of 8 TV sets contains 2 black and white sets and

Question

A shipment of 8 TV sets contains 2 black and white sets and 6 color sets. If 2 TV sets are to be chosen at random from this shipment, what is the probability that at least 1 of the 2 sets chosen will be a black and white set?

A. 1/7
B. 1/4
C. 5/14
D. 11/28
E. 13/28

A. 1/7
B. 1/4
C. 5/14
D. 11/28
E. 13/28

my work:

1 - no B/W (or picking two colors) = 
first pick = 6 / 8
second pick = 5 / 7 
1 - (6/8)(5/7) =
1 - 15/28 =  13/28

the answer given is E) 13/26

Bunuel · Accepted Answer

Let's count the opposite probability of none of the 2 sets will be black-and-white (which means both will be color sets) and subtract this value from 1.

1-\frac{6}{8}*\frac{5}{7}=\frac{26}{56}=\frac{13}{28} .

Same method but using the combinations:  1-\frac{6C2}{8C2}=\frac{13}{28} .

Direct approach: probability of at least 1 set out of 2 will be black-and-white set, is the sum of two probabilities: 1 set b/w and another color plus both sets b/w.

2*\frac{2}{8}*\frac{6}{7}+\frac{2}{8}*\frac{1}{7}=\frac{13}{28} , we are multiplying first fraction by two as b/w and C can occur in two ways: (b/w)(C) and (C)(b/w), first b/w second color or first color second b/w.

Same method but using the combinations:  \frac{2C2+2C1*6C1}{8C2}=\frac{13}{28} .

KillerSquirrel · Answer

1 - the probability for color both times. 1 - 6/8*5/7 = 26/56 = 13/28 Your work is correct !

tt11234 · Answer

why do you times 2? does it mean the order here matters? please advise! thanks~

Bunuel · Answer

I guess you are talking about the red part above.

We can choose 2 televisions of different colors in two ways: (first-b/w)(then-C)=2/8*6/7 or (first-C)(then-b/w)=6/8*2/7 so the probability of this event is the sum of these probabilities --> 2/8*6/7+6/8*2/7=2*2/8*6/7.

Bunuel · Answer

Corrected the answer choices: E should read 13/28.

PROBABILITY APPROACH:

P(at leas one)=1-P(none)=1-6/8*5/7=13/28.

COMBINATIONS APPROACH:

P(at \ leas \ one)=1-P(none)=1-\frac{C^2_6}{C^2_8}=\frac{13}{28} .

Answer: E.

Joy111 · Answer

please clear my understanding

at least one means - either one is black  OR  Both are black

so p( one is black)= 2/8*6/7 = 3/14

P ( Both are black) = 2/8*1/7 =1/28

so p ( at least one is black ) = 3/14 + 1/28 = 1/4 = Answer B

why is this answer differing when we solve it in the alternate way 
 1-p( Both color) = P ( at least one black and white )

Bunuel · Answer

The probability that from 2 sets selected exactly one set is black is  P=2*\frac{2}{8}*\frac{6}{7}=\frac{3}{7} . We are multiplying by 2 since we can select one blacks set in two ways: {Black, Color} or {Color, Black}.

Hope it's clear.

TeamGMATIFY · Answer

Always remember, whenever you see a probability question with atleast 1 in it, find out the probability of none and subtract it from 1

Probability of 0 black and white TV sets = 6C2/8C2

Hence probability of atleast one = 1- 6C2/8C2 = 13/28 Option E

AkshdeepS · Answer

Probability approach:

1. Direct probability:

\frac{2}{8} * \frac{6}{7} + \frac{6}{8} * \frac{2}{7} + \frac{2}{8} * \frac{1}{7}  = \frac{13}{38}

2. Opposite probability approach: (Best for "at least" type questions)

\frac{6}{8} * \frac{5}{7}  = \frac{30}{56}  =  \frac{15}{28} 
   P = 1 -  (None)  =  1-  \frac{15}{28}  = \frac{13}{28}

Combinatorics approach:

1. Direct approach

\frac{(2C1*6C1+2C2)}{8C2}  =   \frac{13}{28}  
 
  2. Opposite combinatorics approach: 
 
  \frac{6C2}{8C2}  =  15/28

P = 1-  \frac{15}{28}  =  \frac{13}{28} 
 
  Answer is E

ScottTargetTestPrep · Answer

The probability that at least 1 of the 2 sets chosen will be a black-and-white set is the probability that exactly 1 set is black and white PLUS the probability that both sets are black and white.

Let’s determine the probability that exactly 1 set is black and white:

The number of ways to choose 2 TV sets from 8 is:

8C2 = 8!/2! (8 - 2) ! = (8 x 7)/(2 x 1) = 56/2 = 28

The number of ways to choose 1 black-and-white TV set (from 2) and thus 1 color TV set (from 6) is:

2C1 x 6C1 = 2 x 6 = 12

Thus, the probability that exactly 1 set is black and white is 12/28.

Now let’s determine the probability that both sets are black and white:

The number of ways to choose 2 TV sets from 8 is still 28.

The number of ways to choose 2 black-and-white TV sets (from 2) and thus 0 color TV sets (from 6) is:

2C2 x 6C0 = 1 x 1 = 1

Thus, the probability that both sets are black and white is 1/28.

Finally, the probability that at least 1 of the 2 sets chosen will be black and white is 12/28 + 1/28 = 13/28.

Alternate Solution:

To find the probability of choosing at least 1 black-and-white set, we can find the probability of choosing no black and white sets and subtract that probability from 1, since our choice will either contain at least 1 black-and-white set or no black-and-white sets).

The number of ways to choose 2 TV sets from 8 is 8C2 = (8 x 7)/(2 x 1) = 56/2 = 28.

The number of ways to choose 2 color TV sets is 6C2 = (6 x 5)/(2 x 1) = 15.

Thus, the probability of choosing no black-and-white sets is 15/28, and so the probability of choosing at least 1 black-and-white set is 1 - 15/28 = 13/28.

Answer: E

ScottTargetTestPrep · Answer

We can use the formula:

P(at least one black and white TV) = 1 - P(No black and white TV)

P(No black and white TV) = 6/8 x 5/7 = 3/4 x 5/7 = 15/28

P(at least one black and white TV) = 1 - 15/28 = 13/28

Answer: E

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A shipment of 8 TV sets contains 2 black and white sets and

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