Applying binomial probabilities

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Daralis

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Apr 29, 2011
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Hey all I've been having a lot of trouble trying to figure this out.

My question is:

It is estimated that only 1% of homes have solar panels. Consider a random sample of 20 homes.

a) what is the probability that none of them have solar panels to generate electricity?

My attempt so far:

n=20
p=0.01

x~bin(20,0.99)

that bit I believe is right

I'm using .99 as that is the percentage of houses that won't be using solar panels. so next do I do this?

P(Y? .99)

???

b) what is the probability that at least one of them has solar panels to generate electricity?

what I think so far:

so for this one

n=20
p=0.01

because the probability as stated is only 1%.

once again I'm stuck.
any pointers would be very much appreciated as I'm studying for an exam which is in 2 weeks!!
 
Use the binomial formula:

\(\displaystyle \binom{n}{k}\cdot p^{k}\cdot (1-p)^{n-k}\)

Where n=20, p=.01, k=0

For the second part, just subtract the above result from 1.

When dealing with an "at least one" problem it is usually best to find the probability of none, then subtract from 1.
 
Ok, I tried it out, I just want to check that I got it right as I don't have the answers, is it:

a) 1.2748

b) .2748

Much appreciated if someone could double check me :)
 
No. You have the first one greater than 1.

You can not have a probability greater than 1. That should be a red flag.

Afterall, we can go from a 0% probability to a 100% probability. Thus, 0 to 1

You are saying there is a 127.48% probability that none have solar panels.

Try this:

\(\displaystyle \binom{20}{0}(.01)^{0}(.99)^{20}\)
 
Yes I was confused when I got over 1. Thank you.

I got from what you put

0.16358

so would that be the probability that one of them has solar panels?
 
I got from what you put 0.16358
Click to expand...

No, sorry, still incorrect.

The probability of NO solar panels is \(\displaystyle \binom{20}{0}(.01)^{0}(.99)^{20}=1\cdot 1\cdot (.99)^{20}\)

so would that be the probability that AT LEAST one of them has solar panels?
Click to expand...

1 minus the above value: \(\displaystyle 1-(.99)^{20}\)

See, "at least one" means there is a bare minimum of 1 that have solar panels up to the max of 20 having solar panels. That there is something. Something being the opposite of nothing :)

So, wouldn't this be the opposite of NONE?. To find the probability of AT LEAST ONE, this is why we subtract the probability of NONE from 1.
 
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