Five mark stop game: why the starting point matters

A quiet paper game can start with five marks and one prediction: if left and right are equally likely, which end will be reached first? The answer depends on the starting mark. In this fictional five mark stop game, a start at one reaches zero with probability three quarters and four with probability one quarter. Its average duration is three steps, although a permitted path can take much longer. The endpoint chance and the average stopping time answer different questions.

Draw or imagine marks zero, one, two, three and four in a line. Start at one. At each interior mark, the next direction is independently left or right with probability one half each, and the position moves exactly one mark in that direction. Stop immediately on reaching zero or four. These are stipulated mathematical rules, not observations about a physical coin, die, app or decorative object. No particular purchase is needed to discuss them.

Write the complete rule before making a prediction

The marks are positions, and a step is one transition between neighboring positions. The starting position is recorded before any step occurs. For example, the path one to zero has two recorded positions but one step. It stops at zero, so no later direction is used in that run. A new start would be another run with its own record, rather than a continuation after an endpoint that the rules already made final.

For a shared afternoon worksheet, write a prediction beside start one before calculating. Then ask the same question for start two. This is an invitation to compare two defined cases, without a claim about anyone's intuition or learning result. A paper discussion can use explicitly imagined direction sequences; the probability calculation comes from the declared independent half-and-half model. Handwritten examples do not establish that a real method of choosing directions has those properties.

Several paths help establish what the rules permit. One to two to three to four ends on the right after three steps. One to two to one to zero ends on the left after three. One to two to one to two to one to zero takes five steps. Every transition moves one mark, and each path stops at its first endpoint. We have not assigned these listed paths equal weights or claimed that they exhaust the possibilities.

Give each starting point an endpoint question

Let u at a position mean the probability of reaching four before zero when starting there. At zero, that probability is zero: the run has already stopped at the left endpoint. At four, it is one because the run is already at the right endpoint. These boundary values supply part of the problem, rather than an instruction to take another step from an endpoint.

From one, a left step stops at zero and a right step reaches two. Each first direction has probability one half. The right-end probability from one is therefore half the right-end probability from two. From three, a left step reaches two and a right step finishes at four, so its probability is half the probability from two plus one half. From two, the first step reaches one or three, making its probability the average of those two probabilities.

Write those three relationships together: u at one equals u at two divided by two; u at three equals u at two divided by two plus one half; and u at two equals the average of u at one and u at three. Each equation conditions on the first step and then retains the same rules for the remaining path. No value is inferred from the appearance of an actual object or from the number of example traces on the page.

Solve the three connected probabilities

Substitute the first and third-position expressions into the middle equation. The result says u at two equals one half of u at two plus one quarter. Subtracting the half leaves one half of u at two equal to one quarter, so u at two is one half. Returning to the other equations gives u at one equal to one quarter and u at three equal to three quarters.

Thus the right-end chances for starts one, two and three are one quarter, one half and three quarters. In particular, equal chances for each direction do not imply equal chances for the two endpoints from start one. That start is adjacent to zero, while reaching four requires surviving later choices. The probabilities follow the entire stopped path. The next direction alone does not have to favor either side for the endpoint chances to differ.

To identify the left-end chance as the complement of the right-end chance, we must also know that the model eventually stops with probability one. An infinitely alternating sequence can be written using the permitted directions, so merely inspecting allowed moves does not prove a fixed deadline. We can instead bound the probability of still being inside after successive blocks of steps.

Explain eventual stopping without inventing a maximum

Imagine the independent direction choices presampled in blocks of three. From any interior mark, three consecutive left choices would reach zero within that block, unless an endpoint had already been reached earlier. The probability of all three being left is one eighth. Consequently, conditional on still being inside at a block's beginning, the chance of surviving its next three steps is at most seven eighths.

After k such blocks, the probability of still not having stopped is at most seven eighths multiplied by itself k times. This bound tends to zero as more blocks are included. Therefore the model stops with probability one. It also has a finite mean duration: grouping survival chances into three-step blocks bounds their sum by three times the convergent geometric sum one plus seven eighths plus seven eighths squared and onward. This is a finiteness argument, not the exact mean we will calculate next.

For the original start one, left and right endpoints now account for the whole probability. The chance of stopping at zero is one minus one quarter, or three quarters. Neither number predicts a required quota in a small series of imagined or observed runs. We have conducted no physical trial. The statement describes the probabilities assigned by this exact model, including its start and its first-endpoint stopping rule.

Ask how many steps remain

Let t at a position be the expected number of further steps until the first endpoint. At zero and four it is zero, because those starts are already stopped. At an interior position, the first step is always counted once. After it, the remaining expected number is the average of the two neighboring values. This gives t at one equal to one plus half of t at two, and the same expression for t at three.

At two, the equation is one plus the average of t at one and t at three. The two outer interior values are equal by their equations, so t at two equals one plus t at one. Substitute that into the first equation. T at one equals one plus half the sum of one and t at one. Multiplying by two and subtracting t at one gives t at one equal to three. Hence t at two is four, and t at three is three.

The mean durations for starts one, two and three are therefore three, four and three steps. Starting at the middle makes the endpoint chances equal and gives the largest of these three means. The units are steps, not seconds or minutes. Nothing in the rules assigns a physical time per step, and the mean does not tell a host exactly when a real gathering or activity will finish.

Keep an average separate from a path limit

From start one, the one-step path straight to zero is allowed. So are the three-step paths already listed and the five-step alternating path. Repeating the one-to-two-to-one segment more times before moving left to zero produces arbitrarily long finite paths. Thus a mean of three does not impose a maximum of three, and the eventual-stopping proof does not give a universal finite deadline for every possible direction sequence.

Changing the rules would require a new question. If someone caps a run at three steps, an unfinished interior position needs a stated outcome; the original model did not define that as an endpoint. If the next direction depends on earlier directions, the independent half-and-half equations may no longer apply. Likewise, changing the starting mark changes which of the calculated values is relevant. Keep such versions labeled rather than mixing their traces into a claim about the original game.

A useful finished worksheet keeps two rows beside each interior start: the chance of ending at four and the mean number of steps. For start one, those entries are one quarter and three; for start two, one half and four; for start three, three quarters and three. The first row measures an event probability and the second a duration average. Label each number with its question and units.

Keep neighboring models and vase accounts separate

The two dice totals guide, count outcomes before totals, maps thirty-six equal-weight ordered pairs onto sums. This stop game instead conditions on the current position and has paths of different lengths. Its endpoint probability is not obtained by counting a short list of paths as equally likely.

The repeated label puzzle, count each repeated-label outcome once, uses four fixed independent draws and separates an any-repeat event from overlapping pair counts. This worksheet stops at a variable first-endpoint time. Neither that fixed-draw complement nor its pair-count average supplies the stopping equations here.

The CosmoLiving account, White Ceramic Handmade Decorative Vase, shows a complete light speckled vase across five images. Flowers, greenery, an elephant figurine, books and room furnishings in its setting are unconfirmed supplied contents. Maker, handmade and material claims are not authenticated, and no dimensions, water holding or performance are tested here.

The Posh and Polish account, White Ceramic Vase with modern tabletop design, shows pierced vessels with matching removable lids across seven images. Its description offers three sizes; pictured multiples do not establish a supplied set or selected size. Graphic dimensions remain attributed, while flowers and surrounding decor are unconfirmed supplies. No material, water holding or performance is established.

Keep a purchase outside the paper game

If you separately consider buying a decorative item, assess the actual offer and current order terms before choosing a payment method. The marks, probabilities and step counts in this worksheet are invented model quantities, not product prices, cryptocurrency amounts or merchant instructions. A completed paper calculation establishes no supplied quantity, material, size or checkout option. For a purchase on republeq.com, inspect the methods presented for the specific order. If cryptocurrency is offered and you consider USDT, confirm the exact asset, network, amount, receiving details, fees and any time limit through that checkout. A familiar token name does not determine its network or settle the total that must be sent. Do not reuse receiving details from a previous basket or infer them from an image, an article or the direction arrows on this worksheet. Review the selected variant, intended quantity, shipping destination, delivery estimate and final total. Resolve any uncertainty about what is actually supplied before authorizing payment. Ask how cancellation, returns and an approved refund would be handled, including the refund asset or currency, any relevant network, calculation basis and possible deductions. Keep those answers with the item and order they concern. Information about another size or a prior purchase may not describe a changed selection. If another person is helping, distinguish a request for product information from authorization to buy, and clarify proposed substitutions. Retain the product reference, order confirmation and any transaction record together. Where an essential instruction remains missing or conflicting, pause and seek clarification through the responsible service. This paragraph is conditional purchasing guidance, not a report of a completed transaction, a promise of current payment availability or advice to select a particular asset. The paper game needs no purchase, and any actual order requires its own present evidence.

For the original start one, save the endpoint chances three quarters left and one quarter right beside the mean of three steps. Keep the five-mark rule, independent half-and-half directions and first-endpoint stop visible. The equal middle-start chances and its mean of four provide a separate control. Those records let another reader check what changed when the starting point changed, without confusing a fair next direction with an equal final chance or a mean with a deadline.