# maths 1B last minute questions (1 Viewer)

#### Drsoccerball

##### Well-Known Member
$\bg_white You are given P(B) = 0.65, P(B|A) = 0.95, P(B^c|A^c) = 0.85.$

$\bg_white Find P(A).$

#### InteGrand

##### Well-Known Member
$\bg_white You are given P(B) = 0.65, P(B|A) = 0.95, P(B^c|A^c) = 0.85.$

$\bg_white Find P(A).$
$\bg_white \noindent We have \mathbb{P}\left(B\mid A^{c}\right) = 1 - \mathbb{P}\left(B^{c}\mid A^{c}\right) = 1 - 0.85 = 0.15. Now,$

\bg_white \begin{align*}\mathbb{P}\left(B\right) &= \mathbb{P}\left(B\mid A\right)\mathbb{P}\left(A\right) + \mathbb{P}\left(B\mid A^{c}\right)\mathbb{P}\left(A^{c}\right) \quad (\text{Law of Total Probability})\\ \Rightarrow 0.65 &= 0.95x + 0.15 \left(1-x\right) \quad (\text{letting }x = \mathbb{P}(A))\\ \Rightarrow 0.5 &= 0.8x \\ \Rightarrow x &= \mathbb{P}(A) = \frac{5}{8} = 0.625.\end{align*}

#### Drsoccerball

##### Well-Known Member
We roll a die successively until we roll six consecutive 6's, in which case we stop.
a) What is the probability that we stop after the thirteenth throw?
b) What is the probability that we roll the die at least 10 times?

#### Drsoccerball

##### Well-Known Member
Thanks guys just a question from the exam:

$\bg_white Suppose p_n denotes the nth prime number.$

$\bg_white Do either of the series \sum_{n=3}^{\infty}{\frac{1}{p_n}} and \sum_{n=3}^{\infty}{(-1)^n\frac{1}{p_n}}$

I said the first one diverges by comparison test and by Leibniz test the second one conditionally converges. Am I right?

#### seanieg89

##### Well-Known Member
Thanks guys just a question from the exam:

$\bg_white Suppose p_n denotes the nth prime number.$

$\bg_white Do either of the series \sum_{n=3}^{\infty}{\frac{1}{p_n}} and \sum_{n=3}^{\infty}{(-1)^n\frac{1}{p_n}}$

I said the first one diverges by comparison test and by Leibniz test the second one conditionally converges. Am I right?
Yes, this is correct. What did you compare the first series to though?

#### Drsoccerball

##### Well-Known Member
Yes, this is correct. What did you compare the first series to though?
I had like a few seconds so i just checked it with the harmonic series without checking the inequality sign... R.I.P

#### Drsoccerball

##### Well-Known Member
Okay it appears I may have read the question wrong :

$\bg_white Let p_n for n \geq 2 denote the number of integers less than or equal to n that are prime. Do either of the series \sum_{n=3}^{\infty}{\frac{1}{p_n}} and \sum_{n=3}^{\infty}{(-1)^n\frac{1}{p_n}} converge?$

#### InteGrand

##### Well-Known Member
I had like a few seconds so i just checked it with the harmonic series without checking the inequality sign... R.I.P
Here's the Wikipedia page for the divergence of the sum of reciprocals of primes (which has a few proofs of it there), if you (or anyone else) want(s) to see it: https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes .

Did they get you to derive some bounds on p_n or something? It's not exactly a trivial thing to prove, that divergence.

#### seanieg89

##### Well-Known Member
Using p_n for pi(n) is horrific notation, but it remains true that the first one diverges and the second one converges, and it remains true that proving divergence is nontrivial (quite similar though).

#### InteGrand

##### Well-Known Member
Using p_n for pi(n) is horrific notation, but it remains true that the first one diverges and the second one converges, and it remains true that proving divergence is nontrivial (quite similar though).
Hahaha was it π(n) that the Q. was actually referring to?

Edit: oh, just noticed Drsoccerball's post above. (Wasn't able to see the LaTeX before, I think it's playing up for me at the moment.)

Did the Q. provide any fact about the asymptotic nature of π(n)?

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#### Drsoccerball

##### Well-Known Member
Hahaha was it π(n) that the Q. was actually referring to?

Edit: oh, just noticed Drsoccerball's post above. (Wasn't able to see the LaTeX before, I think it's playing up for me at the moment.)

Did the Q. provide any fact about the asymptotic nature of π(n)?
What I wrote was what the queation gave

#### seanieg89

##### Well-Known Member
Did the Q. provide any fact about the asymptotic nature of π(n)?
From the wording, they probably just wanted you do decide whether:

a) Neither of them converge.
or
b) At least one of them converges.

which tests students knowledge of the alternating series test whilst also leaving the red herring of the reciprocal prime series (less subject to elementary analysis) to catch out students who are not thinking critically.

#### Drsoccerball

##### Well-Known Member
For those who don't know I got 94 as my mark thanks guys