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

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

Pleaselog inorregisterto add a comment.

How many steps it will take to get one increase in friendship is ageometric 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/pwherepis 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 tox/p, wherexis how many successes you want andpis 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 aredocumented 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.

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

All of this is obviously random. If you have good luck, then you might get a friendship boost everynsteps, 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.