André Nicolas
Reputation
320,830
398/400 score
 1h comment Simple Probability of Playing Cards I assume we are drawing without replacement. Our probability is $\Pr(A\cap B)/\Pr(B)$ where $B$ is the event at least one Ace, and $A$ is the event two Aces. The probability of at least one Ace is $1$ minus the probability of $2$ Kings. The probability of two Kings is $\binom{4}{2}/\binom{8}{2}$. The rest is yours to do. 2h comment the blood test riddle (number theory) You are welcome. The same strategy with $8$ vials would take care of $256$ samples. 2h comment the blood test riddle (number theory) (cont) tells us the ID number of the sample that has the disease $X$. (Added) It is the same for $100$ except we use $7$ bits, since every number from $0$ to $99$, indeed to $127$, has a $7$-bit binary representation. 2h comment the blood test riddle (number theory) Suppose we have $13$ samples, not $100$. Then each sample has a $4$-bit binary ID, Into Vial $1$ we put a little of samples $1,3,5,7,11,13$ (these are the numbers with leftmost bit equal to $1$). Into Vial $2$ we put a little of samples $2,3,6,7,10,11$ (these have second bit from the left equal to $1$). Into Vial $3$ we put a little from samples $4,5,7, 12,13$ and into Vial $4$ a little from $8,9,10,11,12,13$. Now find out which of these test positive. In the $i$-th position from the left, write $1$ if Vial $i$ tests positive, $0$ if it tests negative. The binary number we get (more) 2h comment the blood test riddle (number theory) Hint: Number the samples $0$ to $99$. Into vial $i$, put the blood samples whose ID number has a $1$ in the $i$-th binary position. (So for example Vial $1$ has all the samples whose ID number as a $1$ in the first position, counting from the left, that is, all samples whose ID number is odd.) 12h comment Poisson Distribution Worded Problem (Typist & Corrections Question) I do not know what is intended in your previous comment. The probability that more than $2$ attempts (that is,$3$ or more) are needed is $p^2$. 12h comment Poisson Distribution Worded Problem (Typist & Corrections Question) More than $2$ attempts are needed if the first try had more than one correction and the second try had more than one correction. 12h reviewed Leave Open Is every power of a group element also contained in the group? 12h reviewed Reopen What is the equation to evenly distribute circles in a spiral? 12h reviewed Reopen Diophantine Equations : Solving $a^2$ $+$ $b^2$ $=$ $2c^2$ 12h comment Poisson Distribution Worded Problem (Typist & Corrections Question) Let $p$ be the probability in (a) (I have not checked whether you are right), Then the answer to (b), assuming independence, is $p^2$. 12h comment Questions on Indefinite Integration 2 It is a standard "trick." If we have the integral of $\sin^m x\cos^n x$ where $n$ is odd, we express the $\cos^{n-1}x$ in terms of sines, keeping one $\cos x$ "outside" and then let $u=\sin x$. A similar trick works when $m$ is odd. 13h comment Questions on Indefinite Integration 2 For the first, note that $\cos 2x=1-2\sin^2 x$, so $1-\cos 2x=2\sin^2 x$. We therefore want to integrate $8\sin^6 x\cos^5 x$. Rewrite $\cos^5 x$ as $(1-\sin^2 x)^2\cos x$, and make the substitution $u=\sin x$. 13h comment Questions on Indefinite Integration 2 For the first, let $u=\sin^{-1}y$. Then $du=\frac{1}{\sqrt{1-y^2}}\,dy$, so we want $\int \frac{du}{u}$. 13h comment In any integral domain, only $1$ and $-1$ are their own multiplicative inverses. The first step is to show that $x=x^{-1}$ if and only if $x^2=1$. Then since we are in an integral domain, the product $(x-1)(x+1)$ is equal to $0$ if and only if $x-1=0$ or $x+1=0$, and we are finished. 13h answered Prove that the expected value of a pareto random variable $X$ is equal to$(\frac {a}{a-1})\cdot \lambda$ 1d comment Proving convergence/divergence via the ratio test Sure, I wrote down the reciprocal because the limit for that is more familiar. So the limit of the ratio is $3/e$, bad news for convergence. 1d comment Find the sum of a series An $n$ in the denominator can signal we should integrate something, just like an $n$ in the numerator can indicate we should differentiate something. 1d comment Proving convergence/divergence via the ratio test Or maybe divergence. Note that $\left(\frac{k+1}{k}\right)^k$ has limit $e\lt 3$. 1d comment Find the sum of a series Note that the $n$-th term is $\int_0^z t^{2n-1}\,dt$. So our series is obtained by integrating the geometric series $\sum t^{2n-1}$ term by term.