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having trouble with probability

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annaoh_2003:
im doing an intensive maths course as a prerequisite to uni- but im finding 2nd week a little difficult. I cant keep up with all the formulas and everything. can someone explain how to work this out - do I use a tree diagram or a formula ???

Rose34:

--- Quote from: annaoh_2003 on January 19, 2022, 11:24:22 pm ---im doing an intensive maths course as a prerequisite to uni- but im finding 2nd week a little difficult. I cant keep up with all the formulas and everything. can someone explain how to work this out - do I use a tree diagram or a formula ???

--- End quote ---

Hello,
I tried the question so here is the solution(I could be wrong but here is what I did)
So this whole question is about binomial distribution because you have a sample and you are choosing people from it.

a) Using calculator(I used CAS) "4 people select 3 or more" meaning total number of sample is 4, you select 3 or 4 people from it that have the gene, probability that those people have the gene is 3/5. So using the calculator you select
binomial Cdf(as you have 3 or more people so its kind of an interval)
n=4, p=3/5, lower bound=3, upper bound=4(You can use the formula of biomial distribution but using calculator is faster)
This gives an answer of 0.4752.

b) this is conditional probability because of the phrase "given that". So the conditional probability has the formula of A intersection b divided by probability of B. Here A is the probability that "exactly 2" have the gene, and B is the probability that "at least 1" have the  gene. So the intersection between those 2 is event A which is probability of having "exactly 2"  divide that by the probability of having "at least 1".
Using your calculator:
For the numerator you use biomial pdf (bc of the phrase "exactly 2") thus n=4, p=3/5, X value=2 which gives an answer of 0.3456
For the denominator you use binomial cdf "bc of the phrase "at least 1" thus n=4, p=3/5, lower bound=1, upper bound=4 which gives an answer of 0.9744

Lastly divide A intersection B so 0.3456/0.9744 which gives 0.34568

Again not sure if I am right but that's my answer.

fun_jirachi:

--- Quote from: annaoh_2003 on January 19, 2022, 11:24:22 pm ---im doing an intensive maths course as a prerequisite to uni- but im finding 2nd week a little difficult. I cant keep up with all the formulas and everything. can someone explain how to work this out - do I use a tree diagram or a formula ???

--- End quote ---

Adding onto the above, you can either use a tree or a formula, but a tree with 4 levels seems a little unnecessary.

a) \(P(X \geq 3) = \binom{4}{3}\left(\frac{3}{5}\right)^3\left(\frac{2}{5}\right)^1 +  \binom{4}{4}\left(\frac{3}{5}\right)^4\left(\frac{2}{5}\right)^0\). (Question for you: why does this work, and why is this the same as the calculator answer given above)

b) \(P(X = 2 | X \geq 1) = \frac{P(X = 2)P(X\geq 1 | X = 2)}{P(X\geq 1)} = \frac{P(X=2)}{P(X\geq 1)} = \frac{\binom{4}{2}\left(\frac{3}{5}\right)^2\left(\frac{2}{5}\right)^2}{1-\binom{4}{0}\left(\frac{3}{5}\right)^0\left(\frac{2}{5}\right)^4}\) (Refer to Rose's answer for the reasoning, can't fault it).


--- Quote from: Rose34 on January 20, 2022, 11:25:28 am ---Lastly divide A intersection B so 0.3456/0.9744 which gives 0.34568

--- End quote ---

Pretty sure you just typed the numerator again (0.3456/0.9744 is correct though)

annaoh_2003:

--- Quote from: Rose34 on January 20, 2022, 11:25:28 am ---Hello,
I tried the question so here is the solution(I could be wrong but here is what I did)
So this whole question is about binomial distribution because you have a sample and you are choosing people from it.

a) Using calculator(I used CAS) "4 people select 3 or more" meaning total number of sample is 4, you select 3 or 4 people from it that have the gene, probability that those people have the gene is 3/5. So using the calculator you select
binomial Cdf(as you have 3 or more people so its kind of an interval)
n=4, p=3/5, lower bound=3, upper bound=4(You can use the formula of biomial distribution but using calculator is faster)
This gives an answer of 0.4752.

b) this is conditional probability because of the phrase "given that". So the conditional probability has the formula of A intersection b divided by probability of B. Here A is the probability that "exactly 2" have the gene, and B is the probability that "at least 1" have the  gene. So the intersection between those 2 is event A which is probability of having "exactly 2"  divide that by the probability of having "at least 1".
Using your calculator:
For the numerator you use biomial pdf (bc of the phrase "exactly 2") thus n=4, p=3/5, X value=2 which gives an answer of 0.3456
For the denominator you use binomial cdf "bc of the phrase "at least 1" thus n=4, p=3/5, lower bound=1, upper bound=4 which gives an answer of 0.9744

Lastly divide A intersection B so 0.3456/0.9744 which gives 0.34568

Again not sure if I am right but that's my answer.

--- End quote ---

--- Quote from: fun_jirachi on January 20, 2022, 11:49:18 am ---Adding onto the above, you can either use a tree or a formula, but a tree with 4 levels seems a little unnecessary.

a) \(P(X \geq 3) = \binom{4}{3}\left(\frac{3}{5}\right)^3\left(\frac{2}{5}\right)^1 +  \binom{4}{4}\left(\frac{3}{5}\right)^4\left(\frac{2}{5}\right)^0\). (Question for you: why does this work, and why is this the same as the calculator answer given above)

b) \(P(X = 2 | X \geq 1) = \frac{P(X = 2)P(X\geq 1 | X = 2)}{P(X\geq 1)} = \frac{P(X=2)}{P(X\geq 1)} = \frac{\binom{4}{2}\left(\frac{3}{5}\right)^2\left(\frac{2}{5}\right)^2}{1-\binom{4}{0}\left(\frac{3}{5}\right)^0\left(\frac{2}{5}\right)^4}\) (Refer to Rose's answer for the reasoning, can't fault it).

Pretty sure you just typed the numerator again (0.3456/0.9744 is correct though)

--- End quote ---


that's the problem - no calculator use is allowed in this course :/ Im having issues with the symbols. can someone refer me to some good pages for this kind of stuff?

fun_jirachi:

--- Quote from: annaoh_2003 on January 20, 2022, 09:32:37 pm ---
that's the problem - no calculator use is allowed in this course :/ Im having issues with the symbols. can someone refer me to some good pages for this kind of stuff?

--- End quote ---

Fair enough -- which is why I tentatively wanted you to think about why the non-calculator solution I gave matched the calculator solution. Regardless of your access to a calculator, it's important to translate language/logic into maths in this way ;D (some of the non-calculator solutions will be used by the calculator behind the scenes as well, but it won't show the computation)

I'm not sure which symbols you're referring to -- if you could write them down or draw them out I'm more than happy to either direct you elsewhere for a more comprehensive guide or explain the usage myself!

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