probability – is this the correct way to solve

When two,fair, six sided dice are rolled,the probability of rolling numbers on the two dice that add up to 5 is__?(round to the nearest hundredth)

my work:

2 6 sided dice add up to 5

6×2=12
5/12=0.42
is this the correct answer,and if not what is the correct answer to the solution?

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dnd 5e – How do I calculate d20 success probability using the Halfling ‘lucky’ trait with (dis)advantage?

Here is a comprehensive DPR calculator, and here is the mathematics behind it. I’m trying to follow along with the equations.

At the bottom of the second page are formulas for success probability $L$ of a Halfling (who has luck) in normal circumstances and with advantage and disadvantage: $$L = P + frac{1}{20}P,$$ $$L_{adv} = P_{adv} + left(frac{2}{20}(1 – P) – frac{1}{400}right)P,$$ $$L_{dis} = P_{dis} + frac{2}{20}P^2,$$ where:

  • $P$ is the probability of succeeding on any single roll,
  • $P_{adv} = 1 – (1 – P)^2$ is the probability of succeeding with advantage (not failing both rolls), and
  • $P_{dis} = P^2$ is the probability of succeeding with disadvantage (succeeding both rolls).

The $P$s are quite easy to derive, and $L$ is just passing outright OR (rolling a 1 AND THEN passing the reroll): $$L = P + left(frac{1}{20}*Pright).$$ But I’m struggling with deriving $L_{adv}$ and $L_{dis}$. Please can someone show a derivation?

probability distributions – Averaging dice rolls

Dice are commonly rolled in Dungeons and Dragons to decide the outcome of events. One such case is in combat. First, you roll 1d20 (one 20-sided die) to determine wether your attack hits (attack roll). The attack hits if the value of the attack roll is $ge n$, with $n$ depending on the stats of you and your opponent. Then you do a damage roll to determine the damage dealt by your weapon, with the dice you roll depending on the weapon. The question is, how does the average damage dealt depend on $n$? What if I have two weapons, where I would need to roll 2d20, to see if either hit?

In summary:

  1. Roll 1d20 per weapon (1 or 2)
  2. If the value of the roll is $ge n$, the weapon hits
  3. For each weapon that hits, roll 1d$m$, where $m$ depends on the weapon
  4. Deal damage equal to the value of that roll

For a weapon dealing 1d$m$ damage, what is the average damage dealt when the attack roll needs to be $ge n$? What if you have two weapons? Assume both weapons have the same $m$ value.

I know the raw average of a $p$d$q$ roll is $p*(q+1)/2$ (normally distributed), but I don’t know how the first 1d20 roll affects that, given that it effectively causes the damage roll to be 0, thus skewing the distribution.

statistics – Why does the number of possible probability density functions have the cardinality of the continuum?

Wikipedia’s article on parametric statistical models (https://en.wikipedia.org/wiki/Parametric_model) mentions that you could parameterize all probability distributions with a one-dimensional real parameter, since the set of all probability measures & $mathbb{R}$ share the same cardinality.

This fact is mentioned in the cited text (Bickel et al, Efficient and Adaptive Estimation for Semiparametric Models), but not proved or elaborated on.

This is pretty neat to me. (If I’d been forced to guess, I would have guessed the set of possible probability distributions to be bigger, since pdfs are functions $mathbb{R}rightarrowmathbb{R}$, and we’re counting probability distributions that don’t have a density, too. It’s got to be countable additivity constraining the number of possible distributions, but how?)

Where could I go to find a proof of this, or is it straightforward enough to outline in an answer here? Does its proof depend on AC or the continuum hypothesis? We need some kind of condition on the cardinality of the sample space that neither Wikipedia or Bickel mention, right (if it’s too big, then the number of degenerate probability distributions is too big)?

probability distributions – Quantiles of a Levy process

Let $X = { X_t in {bf R}, t geq 0 }$ be a 1-dimensional (real) Levy process. Suppose further that the distribution of $X_t$ is not concentrated on a grid. (This forces the distribution of $X_t$ to have a Lebesgue density).

For a fixed $p in (0,1)$, let $Q_t(p)$ be the quantile function of $X_t$, i.e $$ {bf P}(X_t leq Q_t(p)) = p. $$

For Brownian motion with drift, $X_t = B_t + alpha t$, $Q_t(1/2)$ is a linear function of $t$. Indeed, $Q_t(1/2) = alpha t$. For $p neq 1/2$, $Q_t(p)$ is not a linear function.

Does any other Levy process have this property? In other words, for a general Levy process, can one find a $p$ that makes $Q_t(p)$ a linear function of time?

What is the probability of mining a block by a miner controlling a fraction of hashing power?

Suppose a miner controls α fraction of the hashing power and the chain on which he is trying to mine has the mining rate given by f. How to find the probability of mining a block by the miner?

pr.probability – Probability distribution of sum of squares of sum/difference of uniform random variables

If we pick $k$ uniformly random integers $x_1,dots,x_kin{0,1,dots,2^t-2,2^t-1}$ then what is the probability distribution of the quantities
$$sum_{substack{i,j=1\ileq k}}^n(x_i-x_j)^2$$
$$sum_{substack{i,j=1\ileq k}}^n(x_i+x_j)^2$$ when $k=t^alpha$ at some $alphain(0,infty)$?

For reference sum of squares of normally distributed variables is given by $chi^2$-distribution while sum of uniform random variables is given by Irwin–Hall distribution.

probability – How to get chance of one ball gonna gets in the red zone if 3 balls of 5 fall into red zone have 48% chance

Throw 5 balls, they can fall on a red or blue field. The chance that 3 balls out of 5 fall on the red field = 48%. Calculate what chance of the one ball gets into the red field, so that this condition is fulfilled.

Well, I got answer about 1%, but for 48% it stop works.
How to get chance of one ball gonna fall to red zone if $3$ balls of $5$ fall into red zone have $1%$

probability theory – Variance of chain length in hashtable

I am have hastable with length $m$. Initially its empty. Next $n/2$ unique random numbers $in (0, n)$ being added to it.

What would be variance of chain length when such $n/2$ numbers being added $x$ times?

I am tested it using code with $n=2^{16}$,$m=2^{15}$ (part of md5 was used as index) and got the following results:

enter image description here
But how to calculate these values without testing or what is the probability of value being added into specific index in the table?