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Probability puzzle

Thu, 04 Jun 2026 06:55:24 +0000

Two players start 1 meter away from a target. They simultaneously begin moving towards the target at a same constant speed. If the left player shoots when he is X meters from the target, his shot hit with a probability 1-X. If the right player shoots when he is X meters from the target, his shot hits with a probability [math]1- X^2[/math]. Each player has exactly one bullet and may choose to shoot at any time during the walk. If exactly one player hit the target, that player wins. If both players shoot simultaneously and both hits , then neither player wins. Both are sent back to the starting positions and game starts over. Similarly, if both the players miss the shots, the game starts over from the beginning. The players don't have to shoot at the same time and they can see each other at all times. Assuming both players use optimal strategies, what is the probability that the left player wins?

What is the relationship between the optical cross-section and the physical cross-section of a particle? - Semiconductor surface engineering

Thu, 09 Apr 2026 14:58:34 +0000

Regarding surface engineering, specifically the reflection and scattering of a semiconductor material, I have the following question in response to the problem below.

On a surface with reflectance R = 0.35, a power of 5 x 10⁻⁵ W/µm² is applied. Knowing that the total power scattered by the particle is 5 x 10⁻² W, determine the optical section. What is the relationship between the optical section and the physical section of the particle?

Scattering Section or Optical Section = σ sc
Physical or geometric section = σ geom
Scattering efficiency factor = Q sc

Substrate reflectivity (R): 0.35.
Incident intensity (I): 5*10E-5 w/(μm^2)
Total scattered power (S or Psc): 5*10E-2 W
Determination of the Optical Section σsc = ?

σ sc= Q sc* σ geom

S: Total power of scattered light (measured in Watts).
I: Intensity of incident light (measured in Watts/μm²).

σ sc= (S(spread))/(I (incident)) or σ sc= (P/ I )

σ sc=((5 ×10^(-2) ) W) / (5 ×10^(-5) W/(μm^2 )) = 10^3 μm^2

What I understood:

The irradiance that effectively interacts with the particle (Ief) is the result of modulating the incident power by the reflectance capacity of the substrate. Mathematically, the presence of the substrate defines the light available for scattering.
Surface reflectance (R) = 0.35
Incident Irradiance (I inc) = 5 E-5 W/(µm^2 )

I ef= R I inc
I ef= 0,35 (5 10^(-5) )
I ef= 1,75 10^(-5) W/(µm^2 )

σ opt=P esp/I ef

σ opt=((5 10^(-2) ))/((1.75 * 10^(-5) ) ) = 2857 µm^2

I'm honestly not sure if the concepts I used in the equations are correct. I would appreciate it if someone could correct me and, most importantly, provide me with bibliographic sources to help me delve deeper into the subject.

≤ can mean- till it or till it OR infinity ?

Tue, 24 Feb 2026 18:19:02 +0000

equalto /greater than OR greater than /equal to

Contextually I understand why it's (0,infinity) but > w/ dash under as 2 means aka till it and till it or infinity

x≤1/2 can mean (0,1/2) or (1/2,infinity)

[Math] [Set] De Morgan's Law, Symmetric difference with 3 sets

Sat, 21 Feb 2026 19:03:50 +0000

This doesn't makes sense

If x e (A1 n A2 n 143 n ... ) then x is not the element of at least one of the sets.

This implies if x doesnt belong to intersection of 3 sets that means it doesn't blong to any set.

De Morgan's Law

I think this is false when C⊂B with elements of A U B

If A A B = A A C, then B

similar to

Symmetric difference with 3 sets

In Symmetric difference why is intersection of 3 not included, also what its associative.

Symmetric difference means "unique to" ? So AΔBΔC != (AΔB)ΔC

Cardinal Number of set Union of 4 Sets

I thinkg it should be - n(AUBUCUD)

= [Sum of number of elements in sets A, B, C and D]

- [Sum of number of elements of intersection of sets taken 2 at a time]

- [Sum of no. of elements of intersection of sets taken 3 at a time]

- Elements of intersection of all sets

when building powersets why don't we make combinations with null set ?

Fri, 20 Feb 2026 19:55:42 +0000

It breaks the rule number of elements of a powerset of Set A = 2^n . Also contradicts null set is a subset of everyset. Also some places it says ∅ is a element of every set

2^4 = 16, but my results say 39

Example for set {1,2,3,4}

There can be 5 no. of sets n+1

single element = n

2 element = 10 - {1,2} {1,3} {1,4} {2,3} {2,4} {3,4} {1,∅} {2,∅} (3,∅) (4,∅)

3 elements = 10

4 elemets = 4 =>{1,2,3,4} {1,2,3,∅} {1,2,∅,4} {1,∅,3,4} {∅,2,3,4}

5 element = 5

additional the ∅

[Eletrochem] Help me understand molar conducitvity

Thu, 19 Feb 2026 13:36:54 +0000

1 mol m–3 = 1000(L/m3) × molarity (mol/L)
The 2nd equation seems to contradict the definetion, how is it using both cm and m in a forumal, i am having trouble understabding the conversion factor

[Chem-Applied] Vapour pressure of Pure Liquids A & B are 450 & 700mmHg respect. @350 K . Find comp. of liquid mix, total Vapour pressure is 600mmHg and Vapour phase composition

Tue, 03 Feb 2026 05:24:51 +0000

xA and xB are mole fractions of each components - Vapour-liquid phase ? why is AI saying it's liquid phase composition ?

Henry's Law : p = KH x “partial pressure of as in vapour phase (p) is proportional to ole fraction of gas (x) in solution” and is expressed as:

Raoult’s Law: A=xA pA∘

pi = yi P total - this is the ratio of 2 components in vapour phase

AI Solved -

Given:
p°A = 450 mm Hg
p°B = 700 mm Hg
Total vapour pressure, P = 600 mm Hg

Let xA be the mole fraction of A in the liquid phase.
Then xB = 1 − xA

Using Raoult’s law:
P = xA·p°A + xB·p°B

600 = 450xA + 700(1 − xA)
600 = 700 − 250xA
xA = 0.40

Therefore:
xA = 0.40
xB = 0.60

Partial vapour pressures:
pA = xA·p°A = 0.40 × 450 = 180 mm Hg
pB = xB·p°B = 0.60 × 700 = 420 mm Hg

Vapour phase composition:
yA = pA / P = 180 / 600 = 0.30
yB = pB / P = 420 / 600 = 0.70

Final Answer:

Liquid phase composition:
xA = 0.40
xB = 0.60

Vapour phase composition:
yA = 0.30
yB = 0.70

[Chem-Applied-Solutions] van’t Hoff factor i

Fri, 06 Feb 2026 19:27:44 +0000

How is it flipped in 2nd step ?

What is the Benzene base compound priority list ?

Sat, 13 Dec 2025 16:41:19 +0000

How to choose the base compound ? And start counting from that ? IS there is a prioty list ?

Derive Midpoint Formula

Tue, 09 Dec 2025 00:12:50 +0000

Derive the midpoint formula for points in the xy-plane.

Derive Distance Formula

Tue, 09 Dec 2025 00:10:46 +0000

Derive the distance formula for points in the xy-plane.

Bubbleverse

Sun, 07 Dec 2025 05:44:28 +0000

I’ve spent the last months building a full mechanical cosmology model from scratch — a “Bubbleverse” simulation framework that evolves from simple single-bubble decay all the way to large-scale cosmic foam.

The project now includes:

energy-driven universe decay (Mark 1–2)

multi-bubble competition and collisions (Mark 3 series)

field-driven cosmology with dynamic critical density (Mark 4–6)

large-scale foam structures and emergent behavior (Mark 7A–7D)

machine-learning analysis of thousands of simulated universes (Mark ML)

It’s not standard physics — just a conceptual model — but the full code, figures, phase maps and explanations are available here:

Www.bubbleverse.dk

If anyone is interested in alternative cosmology models, simulation design, or emergent behavior, I’d love feedback or questions!

Very large summation calculator

Fri, 01 Apr 2022 14:59:13 +0000

In hypergeometric distribution, n= population of n elements, r = sample size, n_{1 =}elements recognized as having some defined criteria, n_{2 =}n - n₁ = remaining elements other than n₁ . We seek the probability q_k such that the sample size r contain exactly k recognized elements provided [math]k \geq 0, k \leq n_1 [/math] if n₁ is smaller or [math] k \leq r[/math] if r is smaller. In such a case the probability [math]q_k =\frac{\binom{n_1}{k}\binom{n- n_1}{r - k}}{\binom{n}{r}} \tag {1}[/math]

Now, I want to calculate n=8500, n₁ =1000, r = 1000, k = 0 to 100. Inserting these values in (1), calculation of summation is very difficult. In such case, how can I use normal approximation to binomial distribution to find q_k ?

Do you have any clue or hint?

Alkenes general formula ?

Thu, 23 Oct 2025 18:47:27 +0000

CnH2n-2d ?

[Organic Chemistry] Nomenclature

Thu, 23 Oct 2025 18:41:54 +0000

5-isopropyl

Iso means 1 branch of methyl on 2^nd

Propyl is c2h4 not 6

Sec-butyl is c with 2 other c attache and another fcitonal group attached

Also iso thing

Propy is c3h8 not 6

Here propyl mean C3H5

What is arrow from N to O in Nitrobenze Resonance Structure

Thu, 30 Oct 2025 19:52:32 +0000

Also why there is +ve charge on II ?

Benzene with polysubstituent nomenclature.

Sat, 01 Nov 2025 19:03:14 +0000

Is the book self-contradictory or am I tripping.

Also what is the order of preference for base substituent ?

Pie bond uses 2 p orbitals ?

Thu, 23 Oct 2025 18:52:26 +0000

'You have already read that the pi (π) bond is a weaker bond due to poor sideways overlapping between the two 2p orbitals."

1 pie bond is actually 2 bonds as it need 2 p orbitals

Why Carbon single bond is sp2 hybridized orbitals ?

Thu, 23 Oct 2025 18:54:09 +0000

Title

[Organic Chemistry] Nomenclature

Thu, 23 Oct 2025 18:44:16 +0000

Chloromethyl ?

C3H5 what is that ?

1. Supposed to be CH3Cl which should be Chloromethyl

Ethyl

Propyl

[Organic Chemistry] Explain decarboxylation

Thu, 23 Oct 2025 18:43:28 +0000

What is acetate ? not methyl carboxylic acid
Why is O negatively charged
oxygen has 6 electrons and when bond it's got 7 if it losses 2 what is 5 but the middle part has 7 electrons
oxygen has 6 electrons and when bond it's got 7 if it losses 2 what is 5 but the middle part has 7 electrons
oxygen has 6 in valence
free radicals has single electrons in their orbital in the valence shell, making atom unstable, reactive, shifts electron cloud

Reaction is from

Organic Chem: Dumas method

Thu, 09 Oct 2025 17:19:22 +0000

Is STP supposed to be 0C 20C or 25C ?
I didn't understand anything
What is 760
What is 273

Organic Chem: Nomenclature

Wed, 15 Oct 2025 17:53:07 +0000

5-isopropyl

Iso means 1 branch of methyl on 2^nd

Propyl is c2h4 not 6

Sec-butyl is c with 2 other c attache and another fcitonal group attached

Also iso thing

Propy is c3h8 not 6

Here propyl mean C3H5

What is tetra

What is tert

Chloromethyl ?

C3H5 what is that ?

1. Supposed to be CH3Cl which should be Chloromethyl

Ethyl

Propyl

What is acetate ? not methyl carboxylic acid
Why is O negatively charged
oxygen has 6 electrons and when bond it's got 7 if it losses 2 what is 5 but the middle part has 7 electrons
oxygen has 6 electrons and when bond it's got 7 if it losses 2 what is 5 but the middle part has 7 electrons
oxygen has 6 in valence
free radicals has single electrons in their orbital in the valence shell, making atom unstable, reactive, shifts electron cloud

Reaction is from

Organic chem explain Wurtz reaction.

Wed, 15 Oct 2025 17:58:00 +0000

Redox Reactions: What takes priority in findng which Element is Oxidized - Add/Remove of H/O, Electron transfer, Electronegative/Positive element ?

Tue, 07 Oct 2025 16:04:09 +0000

Problem 7.1

In (i) What takes priority Electron transfer or Electronegative/Positive element ?
In (i) Explain addition of Electronegative/Positive element seen as oxidation
In (iii) How can Na be oxidized when hydrogen is been added to it ?