A box contains 10 apples, 9 of which are red. An apple is drawn from

Question

A box contains 10 apples, 9 of which are red. An apple is drawn from the box and its color is noted before it is eaten. This is done a total of n times, and the probability that a red apple is drawn each time is less than 0.5. What is the smallest possible value of n?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

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KarishmaB · Accepted Answer

Think about it this way:

Probability that a red apple is picked in each pick =  \frac{9}{10}*\frac{8}{9}*\frac{7}{8}*... 
Note that all numerators and denominator will get cancelled except the first denominator 10 and the last numerator. Since this probability < 0.5, the last numerator must be less than 5 i.e. 4. 
Probability that a red apple is picked in each pick =  \frac{9}{10}*\frac{8}{9}*\frac{7}{8}*\frac{6}{7}*\frac{5}{6}*\frac{4}{5} = \frac{4}{10}

So the last pick would be out of 5 apples and hence 5 apples would have already been picked. So the last pick would be the 6th pick.

Answer (D)

Archit3110 · Answer

i used hit and trial to get value
9c6/10c6 = .4
n has to be 6
IMO D

malvika2706 · Answer

I calculated valued when:

-> n=1 , p=9/10 . Total p=0.9
-> n=2 , p=8/9 . Total p=0.9*(8/9) it is greater than 0.5
-> n=3 , p=7/8 . Total p=0.9*(8/9)*(7/8) it is greater than 0.5
-> n=4 , p=6/7 . Total p=0.9*(8/9)*(7/8)*(6/7) it is greater than 0.5
-> n=5, p=5/6 . Total p=0.9*(8/9)*(7/8)*(6/7)*(5/6) it is greater than 0.5
-> n=6, p=4/5, total p = 0.9*(8/9)*(7/8)*(6/7)*(5/6) *(4/5)  this is less than 0.5 
for n>6, total p will continue to decrease
hence smallest possible value of n is 6

Answer: D

metalhead2593 · Answer

P = (9/10)*(8/9)*...*  = (10-n)/10 < 1/2

So 20-2n < 10
n > 10/2 =5.
Minimum value of n = 6.

grassbird · Answer

Technically, when n=5, p= 9/10 x 8/9 x 7/8 x 6/7 x 5/6 = 0.5, not greater than 0.5, but the prompt ask for smaller than 0.5, the answer remains unchanged, I just wanna point out the calcualtion

ScottTargetTestPrep · Answer

Solution: If n = 1, the probability that a red apple was chosen each time is 9/10. If n = 2, the probability that a red apple was chosen each time is 9/10 * 8/9 = 8/10. Following this logic, we see that the probability that a red apple was chosen each time for n times is 9/10 * 8/9 * 7/8 * … * (10 - n)/(11 - n). This expression simplifies to (10 - n)/10. We are looking for the smallest possible value of n such that (10 - n)/10 is less than 0.5 = 1/2. We can create the following inequality: (10 - n)/10 < 1/2 20 - 2n < 10 2n > 10 n > 5 Thus, the smallest possible value of n is 6. Answer: D

Thekingmaker · Answer

it's quite easy yhe probability goes as following 9*8*7*6*5*4/10*9*8*7*6*5 = 0.4 this is when the probability goes down below 0.5 this has to be done 6 times therefoew IMO D

MathRevolution · Answer

P(red) =  \frac{9}{10} = 1^{st}  time

P(red) =   \frac{9}{10} * \frac{8 }{ 9} = \frac{8}{10} = 2^{nd}  time

=> For 5 times, it will reach to  \frac{5}{10} = 0.5.

=> We need the probability to be less than 0.5. Hence, n > 5 which is 6

Answer D

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A box contains 10 apples, 9 of which are red. An apple is drawn from

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A box contains 10 apples, 9 of which are red. An apple is drawn from

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