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The Bulbapedia article here mentions the probabilities for GSC, but not HGSS. What is the probability in HGSS to get a hair cut from both the brothers?

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As sumwun said in the other answer, there is 100% probability to get a haircut. But we need the probability of how much the friendship/beauty etc are raised, which isn't in Bulbapedia, if that is what you mean.
Really? No, you are wrong. I need a source for to verify. Probability by which you have the chance of getting a haircut. I'm sure if it was 100% then Bulbapedia would have done it already.
According to my HgSs guidebook that I accidentally stole from my elementary school library nearly a decade ago (long story), it says the following:

Sunday: Junior Stylist
Monday: N/A
Tuesday:  Senior Stylist
Wednesday:  Junior Stylist
Thursday: Senior Stylist
Friday: Junior Stylist
Saturday: Senior Stylist
There are no days where the two overlap, and the only day you cannot get a haircut is Monday.

And according to the source you provided earlier, it says "The younger of the Haircut brothers is cheaper at $300, but his service is often less satisfactory. However, the younger brother has a chance of occasionally increasing a Pokémon's friendship to a greater degree than the older brother is capable. The older brother is more reliable but more expensive at $500."

I believe this means the younger brother has a higher chance of having a positive result, thus raising the friendship more than the older brother, while the other brother has a higher chance of raising the beauty and sheen aspects of the Pokemon.

"Looks a little happier." Gives friendship of +1   
"Looks happy!" Gives friendship of +5   
"Looks delighted!" Gives friendship of +10

For best results, I reccomend you to save before sending your Pokemon to the brothers and resetting until you get the result you want, and it would be more worthwhile to go on the days the younger brother is there.

I unfortunately cannot find the exact percentages you're looking for, but I hope this helps to some degree. I believe it's safe to assume the percentages are similar to that of GSC.

In conclusion, go to the younger hair brother for the best friendship-related results, and soft reset until you got the result you're hoping for.
I'm pretty sure the question is specifically asking for exact percentages. Most of the stuff in this answer is already on Bulbapedia.
Yes sumwun, but if there is no exact percentages out there, then it'll be impossible to answer. As of now, i gave my recommendation of what to do and how to do it, but if someone else finds percentages then bravo to them. In the meantime, hopefully this helps the poster until those percentages are found
TY ™, I am going to upvote it but not BA it so that if somebody finds the percentage, then they could give their answer. Whatever you tell on the answer, 99% is already on that Bulbapedia article.
I think more people would look for exact percentages if they saw this question on the unanswered list.

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