Adam, Bob, and Carol wait in a line, each intending to buy one of the

Question

Adam, Bob, and Carol wait in a line, each intending to buy one of the six hats of different colors: red, blue, yellow, green, black, and white. What is the probability that the blue hat will be bought by Bob or Carol?

A. \(\frac{1}{120}\)

B. \(\frac{1}{24}\)

C. \(\frac{1}{6}\)

D. \(\frac{1}{5}\)

E. \(\frac{1}{3}\)

A.  \frac{1}{120}

B.  \frac{1}{24}

C.  \frac{1}{6}

D.  \frac{1}{5}

E.  \frac{1}{3}

sanjitscorps18 · Accepted Answer

Lets say Bob buys it first then
P(Adam - non-blue hat) x P(Bob - blue hat) x P(Carol - non-blue hat) = 5/6 x 1/5 x 4/4 = 1/6

Similarly, if Carol buys it then we have
P(Adam - non-blue hat) x P(Bob - non-blue hat) x P(Carol - blue hat) = 5/6 x 4/5 x 1/4 = 1/6

Total probability by adding the above = 1/3

IMO Option D

MS05 · Answer

Step 1: Total possibilities
There are 6 hats, and each person can get any hat, without replacement.
 So the total number of possible ways the 3 people can get hats =
 6×5×4=120
 
Step 2: Favorable outcomes
We want cases where  the blue hat  goes to  Bob or Carol .
 
 Case 1: Blue → Bob
 Then Adam and Carol can choose from the remaining 5 hats and 4 hats respectively → 5×4=20 ways. 
 Case 2: Blue → Carol
 Then Adam and Bob can choose from the remaining 5 and 4 hats respectively → 5×4=20 ways.  
Total favorable = 20+20=40
 
Step 3: Probability
P(Blue → Bob or Carol)=40/120=1/3 (E)

ChevyC · Answer

couldn't we just think of it like this:

because it's an "or" it's addition.

P(Bob buys a blue hat) = 1/6

P(Carol buys a blue hat) = 1/6

Therefore, P(bob or carol buys the blue hat) = 1/6+1/6 = 1/3

luisdicampo · Answer

Deconstructing the Question  
Three buyers:  Adam, Bob, Carol .
There are  6 different hats , including blue.
We want the probability that  Bob or Carol gets the blue hat .

Key idea: the blue hat is equally likely to be chosen at any purchase position.

Step-by-step

The blue hat can appear in any of the 6 purchase positions:
 \frac{1}{6}	ext{ each}

Bob and Carol occupy 2 of those positions:
 \frac{2}{6}=\frac{1}{3}

Answer: 1/3

Adam, Bob, and Carol wait in a line, each intending to buy one of the

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Adam, Bob, and Carol wait in a line, each intending to buy one of the

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