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01 Mar 2021, 08:45
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D
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Difficulty:
95%
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Question Stats:
43% (02:13) correct
57%
(02:20)
wrong
based on 125
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The probability that a certain variety of plant produces a yellow flower is 1/5. What is the least number of plants which need to be chosen so that the probability of at least one yellow flower is greater than 0.99?
(A) 18
(B) 19
(C) 20
(D) 21
(E) 22
Are You Up For the Challenge: 700 Level Questions
(A) 18
(B) 19
(C) 20
(D) 21
(E) 22
Are You Up For the Challenge: 700 Level Questions
01 Mar 2021, 17:59
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Bunuel
We may change the question to ".... the probability of none yellow less than 0.01?"
If we choose n plants, the probability of none yellow is \((\frac{4}{5})^n\) and we want that to be less than 0.01.
\((\frac{4}{5})^n < \frac{1}{100}\).
I believe you'll be forced to use a calculator to get the answer for this (if there's a more GMAT way of getting the answer I'll be happy to know!), we input ln(0.01) / ln(0.8) to get n > 20.6 (when we divide both sides by ln(0.8) the signs change since ln(0.8) is negative). So the least n is 21.
Ans: D
01 Mar 2021, 20:34
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P(Yellow) = 1/5
P(Non yellow) = 1 - 1/5 = 4/5
P(At least 1 yellow) = 1 - P(All non yellow)
Let the number of times a flower is picked = n
Then \(1 - (\frac{4}{5})^n > 0.99\)
\((\frac{4}{5})^n < 1 - 0.99 = 0.01 = \frac{1}{100}\)
\(\frac{4}{5} = \frac{8}{10}\)
I took n = 20 as I knew the value of 2^10 = 1024
\((\frac{2^3}{10})^{20} = \frac{(2^{10})^6}{10^{20}}= \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024}{10^4} * 1024\)
= 0.1 * 0.1 * 0.1 * 0.1 * 0.1 * 1024
= 0.01 * 0.01 * 1024 * 0.1
= 0.1024 * .1 = 0.0124
This is a rough approximate, but at n = 20, the value is just greater than 0.01. Therefore n should be 21
We can check by the same method as used above
\((\frac{2^3}{10})^{21} = \frac{(2^{10})^6 \space * \space 2^3}{10^{21}}= \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024 * 8}{10}\)
= 0.1 * 0.1 * 0.1 * 0.1 * 0.1 * 819.2
= 0.01 * 0.01 * 81.92
= 0.01 * 0.8192
= 0.008192
Which is less than 0.01
Option D
Arun Kumar
P(Non yellow) = 1 - 1/5 = 4/5
P(At least 1 yellow) = 1 - P(All non yellow)
Let the number of times a flower is picked = n
Then \(1 - (\frac{4}{5})^n > 0.99\)
\((\frac{4}{5})^n < 1 - 0.99 = 0.01 = \frac{1}{100}\)
\(\frac{4}{5} = \frac{8}{10}\)
I took n = 20 as I knew the value of 2^10 = 1024
\((\frac{2^3}{10})^{20} = \frac{(2^{10})^6}{10^{20}}= \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024}{10^4} * 1024\)
= 0.1 * 0.1 * 0.1 * 0.1 * 0.1 * 1024
= 0.01 * 0.01 * 1024 * 0.1
= 0.1024 * .1 = 0.0124
This is a rough approximate, but at n = 20, the value is just greater than 0.01. Therefore n should be 21
We can check by the same method as used above
\((\frac{2^3}{10})^{21} = \frac{(2^{10})^6 \space * \space 2^3}{10^{21}}= \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024}{10^4} * \frac{1024 * 8}{10}\)
= 0.1 * 0.1 * 0.1 * 0.1 * 0.1 * 819.2
= 0.01 * 0.01 * 81.92
= 0.01 * 0.8192
= 0.008192
Which is less than 0.01
Option D
Arun Kumar
