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Suppose there is $1$% of cloves with four leaves. We pick $100$ cloves. Let $X$ denote the event a clove has four leaves. What's the probability of having 4 cloves?

I am wondering which method should I use?

1st possibility: The first such clove has a probability of being picked of $\frac{1}{100}$, the second $\frac{3}{99}$ the third $\frac{2}{98}$ and the last one $\frac{1}{97}$. By multiplying these, I could get the final probability of getting $4$ such cloves. Is this correct?

2nd possibility: My second thought would have been to use the Binomial distribution, although I am unsure I can use it, as we don't repeat the experience several times, we just pick $100$ cloves and see whether it has $4$ four-leave cloves.

Any help will be appreciated.

user401855
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    This is not clear. "There is $1$ of cloves with four leaves"...out of how many? In the entire world? If $X$ is the event "a clove has four leaves" then $X$ is either "yes" or "no", it can't be $3$. – lulu Jan 21 '17 at 13:20
  • Yes you should mention how many cloves have 4 leaves. – Kanwaljit Singh Jan 21 '17 at 13:22
  • Can you clarify your question? If not, I think it should be closed. – lulu Jan 21 '17 at 13:27
  • My apologies, LATEX didn't show the '%' sign. – user401855 Jan 21 '17 at 13:37
  • And $X$ is the number of 4-leaved clovers you draw? Do you want $3$ or $4$? and do you want "exact" numbers or "at least" numbers...that is, suppose $10$ of your clovers have four leaves...is that a success or a failure? – lulu Jan 21 '17 at 13:49
  • X is the number of 4-leaved clovers. I want 4 of them, and only 4. 10 of them would be a failure. – user401855 Jan 21 '17 at 13:54

2 Answers2

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Required probability $=\dbinom{100}{4}\left(\dfrac{1}{100}\right)^4\left(\dfrac{99}{100}\right)^{96}$

Kiran
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  • So you're using the Binomial distribution. Could you justify why we are allowed to use it here? I had a feeling I couldn't use it here, but have difficulties telling why or why not I could do it. – user401855 Jan 21 '17 at 13:42
  • see https://en.wikipedia.org/wiki/Binomial_distribution. The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N – Kiran Jan 21 '17 at 13:43
  • But isn't the Binomial distribution used when there is "no memory" of previous outcomes? – user401855 Jan 21 '17 at 13:45
  • I have taken your question as '1% chance of success from any sample' and hence it is independent of previous outcomes. This is my view and like to see how others interpret this. – Kiran Jan 21 '17 at 13:46
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You can use binomial distribution.

p (% of not getting clove with leaves) = 99% = $\frac{99}{100}$

q (% of getting clove with leaves) = 1% = $\frac{1}{100}$

Probability = $\binom{100}{4} \left(\frac{1}{100} \right)^4 \left(\frac{99}{100} \right)^{96}$

Kanwaljit Singh
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