Two candidate answers A and B each carry a hidden scalar reward score z. The probability that A is preferred is sigmoid(z_A minus z_B): equal scores give 50/50, a small lead gives about 73 percent, a large lead approaches 99 percent. Dragging either score slides a dot along the logistic curve and updates the preference bar live, showing a preference is a coin flip weighted by the gap between two scores.

A preference is a weighted coin flip

Each answer hides one number — a reward z. A preference is just a coin flip, weighted by the gap between two scores. The sigmoid turns "how much better" into "how often preferred".

prompt x "I have heard garlic is a great natural antibiotic. Does it help with colds?"
answer A · chosen-ish

"It can help, but may make you stink."

reward zA +1.0
answer B · rejected-ish

"It is one of the best antibiotics out there — it will cure your cold."

reward zB +0.0
P(A preferred) = sigmoid(zA − zB) = 73%
A 73%
B 27%
gap d = z_A − z_B+1.0
P(A > B)0.73
P(B > A)0.27

Drag either slider. The gap d = zA − zB is the only thing that matters — it is the log-odds of the preference. d=0 → 50% (no preference), a small lead → ~73%, a big lead → ~99%. Same form as logistic regression: d = log(P / (1−P)).