Circuit to randomly light one LED or another
This is 50-50. 

edit: Above may have Nch-Pch square symmetry error with temp so added divide by 2 and raised clock to 100kHz and removed inverter so it selects random binary phase from LPF filtered rising clock.

Improved Sim with noise injected for test on supply.

Use any button switch to ground.  

Extra LED's with Shift Register to see history.

Clock frequency is so fast compared to human response, it is random.




A leaky cap with 1MΩ resistance will offset the 50-50 by Rf/1MΩ = 1kΩ/1MΩ=0.1%      
10kΩ Series to protect CMOS input.      
Small signal diode to prevent V- on rapid button pushing.    
A good ceramic cap is 50 seconds=ΩF so 100nF = 500 MΩ but then dirt flux is less than this.
Do you feel lucky guessing the next or how many times to get 9 same in a row?