How many steps does one have to take to reach Max Happiness starting at base 0?

Question

I'd like results for when a Pokemon has the Soothe Bell and when it doesn't

Comments

which game?

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Max friendship is 255. Note that steps usually increase friendship by 1, and using a soothe bell will make the game round down and still increase the friendship by 1.

Generation 2: 255 friendship * 512 steps per friendship = 130560 steps
Generations 3, 4: 255 friendship * 128 steps per roll / 50% chance of friendship per roll = 65280 steps
Generation 5: 200 friendship * 128 steps per roll / 50% chance of success per roll / 2 friendship per success + 55 friendship * 128 steps per roll / 50% chance of success per roll / 1 friendship per success = 39680 steps
Generation 5 with soothe bell: 200 friendship * 128 steps per roll / 50% chance of success per roll / (2 \* 3/2) friendship per success + 55 friendship * 128 steps per roll / 50% chance of success per roll / 1 friendship per success = 31147 steps

I couldn't find the exact chance of raising friendship every 128 steps in generations 6 and 7.

Answer 1

How many steps it will take to get one increase in friendship is a geometric random variable, with the caveat that you need to multiply the result by the number of steps it takes to trigger one trial.

The expected value (average) of a geometric random variable is simply 1/p where p is the chance of success on each trial. For example, flipping a coin in order to get heads has an expected value of 1/0.5 = 2. Per the above, if we have a 50% chance of getting +1 friendship every 128 steps, then our expected value is 1/0.5 * 128 = 256 steps. All very simple.

But we want to reach 255 friendship, not 1 friendship. Fortunately, an expected value for this is also very easy to calculate for geometric random variables: you just add the expected number of trials to the first success, plus the expected number of trials to the second success, etc. This simplifies to x/p, where x is how many successes you want and p is the chance of success on every trial.

The only complicating factor is that in most games, walking increases friendship at a different rates depending on the existing friendship. So we need to split our calculations, account for the different boosts, and add the results together. I won't repeat the mechanics of this that are documented on Bulbapedia, only list the calculations we're concerned about.

Of course, there is also per-generation variance in the number of steps per roll and the chance of success on each trial. The chance of success on each trial is missing on Bulbapedia for Gen 6 and 7; I'm just going to assume it's 50%. Take those calculations with my reservations.

Without Soothe Bell

Gen 1: 100/0.5 * 255 / 2 + 155/0.5 * 255 = 104,550 steps average
Gen 2: 255/1 * 512 = 130,560 steps on average
Gen 3: 255/0.5 * 128 = 65,280 steps average
Gen 4: 255/0.5 * 128 = 65,280 steps average
Gen 5: 200/0.5 * 255 / 2 + 55/0.5 * 255 = 79,050 steps on average
Gen 6*: 200/0.5 * 128 / 2 + 55/0.5 * 128 = 39,680 steps on average (*assuming 50%)
Gen 7*: 200/0.5 * 128 / 2 + 55/0.5 * 128 = 39,680 steps on average (*assuming 50%)
LGP/E: 200/0.5 * 255 / 2 + 55/0.5 * 255 = 79,050 steps on average
Sw/Sh: n/a (walking cannot increase friendship to max)
BD/SP: 255/0.5 * 128 = 65,280 steps average
Gen 9: not researched yet

With Soothe Bell

Soothe Bell only applies when the happiness boost per n steps is greater than 1, because Pokemon rounds down. This matters for the +2 boost at low happiness in some games, which becomes +3.

Gen 3: 255/0.5 * 128 = 65,280 steps average
Gen 4: 255/0.5 * 128 = 65,280 steps average
Gen 5: 200/0.5 * 255 / 3 + 55/0.5 * 255 = 62,050 steps on average
Gen 6*: 200/0.5 * 128 / 3 + 55/0.5 * 128 = 31,147 steps on average (*assuming 50%)
Gen 7*: 200/0.5 * 128 / 3 + 55/0.5 * 128 = 31,147 steps on average (*assuming 50%)
Sw/Sh: n/a (walking cannot increase friendship to max)
BD/SP: 255/0.5 * 128 = 65,280 steps average
Gen 9: not researched yet

All of this is obviously random. If you have good luck, then you might get a friendship boost every n steps, in which case the number of steps will be lower than the above. If you have bad luck, then it will be higher. We have only calculated averages.

Comments

This is totally incorrect answer considering it's like you take 128 steps and you have a 50% chance to get +1 and 50% chance to get 0! there is also the chance you get all the 65,280 steps and receive 0 friendship. You can't measure how many steps it will take when it's rng. Period. It's incorrect maths.

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All of the above is addressed throughout the answer, especially in the last paragraph. Maybe you should read it an extra time. You act like probability is an illegitimate branch of maths.