Probability Question! (1 Viewer)

[Kimyia]

I'm terrible at probability and I don't know how to go about answering this question! Can anyone help?

The probability that a plant grows to maturity is 1/9. What is the least number of plants needed to have at least a 95% chance of one plant reaching maturity?

Thank you

 

[Demise]

I remember these questions, I used to love them. I remember that you need to make them a GP.

A1 = 1/9
A2 = (8/9 x 1/9)
A3 = (8/9 x 8/9 x 1/9)
An = [ (8/9)^n-1 x 1.9) ]

A1 + A2 + A3 + ... + An = 0.95

A = 1/9
R = 8/9

0.95 = [1/9(1 - (8/9)^n)] / (1/9)

0.95 = 1 - (8/9)^n

(8/9)^n = 0.05

n ln (8/9) = ln 0.05

n = ln 0.05 / ln (8/9)
n = 25.43
n = 26

Is that right?

 

[deswa1]

I haven't worked it out but just looking at the above answer- it doesn't seem right. Consider that one plant has a 1/9 chance of reaching maturity so if you have 9 plants, its reasonable to expect one to reach maturity (obviously it doesn't work quite like that) but its a good appromixation. 26 then seems too big. OP, is the question what is the minimum to have at least one plant reach maturity or exactly one plant reach maturity? (note sure if the second can be solved actually given this question- I'll see)

 

[Kimyia]

@Demise Yes, you got it right! And thank you, that makes sense!!

@dewsa1 The question said "what is the least number of plants needed for at least a 95% chance of one plant reaching maturity"

 

[Demise]

I haven't worked it out but just looking at the above answer- it doesn't seem right. Consider that one plant has a 1/9 chance of reaching maturity so if you have 9 plants, its reasonable to expect one to reach maturity (obviously it doesn't work quite like that) but its a good appromixation. 26 then seems too big. OP, is the question what is the minimum to have at least one plant reach maturity or exactly one plant reach maturity? (note sure if the second can be solved actually given this question- I'll see)

We thought that too, but it's not assuming that there's 9 plants, it's only saying that the probability is 1/9. 26 doesn't seem too big, because the thing is, it's asking at what point is the chance that the seed planted (NOW) will have a 95% chance of growing? Every single seed planted decreases in chance, so you need to continuously add each scenario until you reach that 95% chance since that's just how it works.

 

[Demise]

@Demise Yes, you got it right! And thank you, that makes sense!!

@dewsa1 The question said "what is the least number of plants needed for at least a 95% chance of one plant reaching maturity"

Anytime man, if you have any more probability questions, hit me up.

 

[Carrotsticks]

I think the question could be made clearer by rephrasing it as such:

"What is the least number of plants needed for at least a 95% chance of EXACTLY one plant reaching maturity"

 

[Demise]

I think the question could be made clearer by rephrasing it as such:

"What is the least number of plants needed for at least a 95% chance of EXACTLY one plant reaching maturity"

That would make sense, but ultimately both questions direct you into figuring it out GP style.