Is a weighted average, assuming the weights sum up to one, always smaller than the unweighted arithmetic average? As if the weights are smaller than one the weighted mean is a convex combination of the individual points. Is there any condition on the individual observation for the statement to hold true?

Is anybody researching "ternary" groups?
As someone who has an undergraduate education in mathematics, but didn't take it any further, I have often wondered something.
Of course mathematicians like to generalize ideas. i.e. it is often better to define and write proofs for a wider scope of objects than for a specific type of object. A kind of "paradox" if you will - the more general your ideas, often the deeper the proofs (quoting a professor).
Anyway I used to often wonder about group theory especially the idea of it being a set with a list of axioms and a binary function $a\cdot b=c$. But has anybody done research on tertiary (or is that trinary/ternary) groups? As in, the same definition of a group, but with a $\cdot (a,b,c)=d$ function.
Is there such a discipline? Perhaps it reduces to standard group theory or triviality and is provably of no interest. But since many results in Finite Groups are very difficult, notably the classification of simple groups, has anybody studied a way to generalize a group in such a way that a classification theorem becomes simpler? As a trite example: Algebra was pretty tricky before the study of imaginary numbers. Or to be even more trite: the Riemann $\zeta$ function wasn't doing much before it was extended to the whole complex plane.
EDIT: Just to expand what I mean. In standard groups there is an operation $\cdot :G\times G\to G$ I am asking about the case with an operation $\cdot :G\times G\times G\to G$

A random sample of 5,000 people is selected from local telephone book to participate in financial planning survey to help infer about typical saving and spending habits. three thousand return the questionnaire. Describe the types of bias that may result in at least 4-5 complete sentences.

Calculating conditional probability given Poisson variable
I encountered a set of problems while studying statistics for research which I have combined to get a broader question. I want to know if this is a solvable problem with enough information specifically under what assumptions or approach.
Given that a minesweeper has encountered exactly 5 landmines in a particular 10 mile stretch, what is the probability that he will encounter exactly 6 landmines on the next 10 mile stretch. (Average number of landmines is 0.6 per mile in the 50 mile stretch)
I have figured that the approach involves finding out the Poisson probabilities of the discrete random variable with the combination of Bayes Conditional probability. But am stuck with proceeding on applying the Bayes rule. i.e $Pr(X=6\mid X=5)$.
I know that $Pr(X=5)=e^{-6}{5}^6/5!.$ Here $\lambda =0.6\cdot 10$ and $X=5$) Similarly for $Pr(X=6)$. Is Bayes rule useful here: $P(Y\mid A)=Pr(A\mid Y)Pr(Y)/(Pr(A\mid Y)Pr(Y)+Pr(A\mid N)Pr(N))$?
Would appreciate any hints on proceeding with these types of formulations for broadening my understanding.

Let $A$ and $B$ be some continuous random variables. We proceed as follows: we take a coin with bias $b$ and flip it. If heads, we inspect $A$, if tails we inspect $B$. Call this resulting random variable $C$.
Now say I can observe $C$ and want to figure out if the coin was heads or tails, i.e. I want to compute $Pr[$
One thought I had was to approximate the coin toss with a continuous random variable $K$ with pdf:
$k(x)=b,\text{ for }x\in [-1,0]$
$k(x)=1-b,\text{ for }x\in [0,1]$
Then one could compute the joint density function for $K$ and $C$ and compute $Pr[K<0|C=c]$ from there. But this looks clunky and ugly. Is there a better way?

How do I determine sample size for a test?
Say you have a die with n number of sides. Assume the die is weighted properly and each side has an equal chance of coming up. How do I determine the minimum number of rolls needed so that results show an equal distribution, within an expected margin of error?
I assume there is a formula for this, but I am not a math person, so I don't know what to look for. I have been searching online, but haven't found the right thing.

Study of an infinite product
During some research, I obtained the following convergent product $P_a(x):=\prod _{j=1}^{\mathrm{\infty }}\cos \left(\frac{x}{j^a}\right)(x\in \mathbb{R},a>1).$.
Considering how I got it, I know it's convergent and continuous at 0 (for any fixed a), but if I look at $P_a$ now, it doesn't seem so obvious for me.
Try: I showed that $P_a\in L^1(\mathbb{R})$, i.e. it is absolutely integrable on $\mathbb{R}$. Indeed, by using the linearization of the cosine function and the inequalities $\ln (1-y)⩽-y$ (for any $y<1$) and $1-\cos z⩾z^2/2$ (for any real z), we obtain
$\begin{array}{rcl}{P}_a(x{)}^2& =& \prod _{j=1}^{\mathrm{\infty }}(1-\frac{1-\cos (x{j}^{-a})}{2})⩽\prod _{j>|x{|}^{1/a}}(1-\frac{1-\cos (x{j}^{-a})}{2})⩽\exp (-C|x{|}^{1/a}),\end{array}$ for some absolute constant $C>0$. However, I have not been able to use this upper bound to prove continuity at 0 (via uniform convergence for example, if we can).
Question : I wanted to know how to study $P_a$ (e.g. its convergence and continuity at 0), if you think it has an other form "without product" (or other nice properties) and finally if anyone has already seen this type of product (in some references/articles), please.

From $10$ numbers $a,b,c,...j$ all sets of $4$ numbers are chosen and their averages computed. Will the average of these averages be equal to the average of the $10$ numbers?