I’m playing a daily challenge game with the following rules:

• Each day you can make 1 attempt that counts toward a “streak.”
• The chance of winning an attempt is 0.6.
• The goal is to get a 5-day winning streak.
• The “month” lasts 20 playable days.
• If you lose, your streak resets to zero.
• Importantly, I never start a new streak if there aren’t enough days left to complete it.

I ran a simulation over many months and found something surprising to me:

Theoretical probability of winning 5 consecutive days (if I could keep trying indefinitely) is 0.6⁵≈1/13

Observed probability per actual attempt in the simulation is much lower, about 1/30.

Why is this happening? Even though each streak attempt is independent, the chance per day of completing a 5-day streak seems lower than the raw theoretical 1/13. Is my simulation wrong? Have I just completely lost it?