Yes or no, one chance in two each.
How the yes or no answer is drawn
Each tap draws one of two outcomes with equal weight, so yes and no each have one chance in two, the same as a fair coin. The question you type is not read by anything. It sits above the answer so you remember what you asked, and rewording it, adding a question mark or asking the same thing five times has no effect on what comes back.
Every answer is independent of the one before. Three noes in a row do not make a yes any more likely on the fourth tap, because the draw has no memory of earlier ones. The chance is one in two every time, however the last few went.
What adding maybe does to the odds
With maybe switched on there are three answers, and each gets one chance in three. That is a bigger change than it looks. Yes falls from 50 percent to about 33, and so does no. Over thirty taps you would expect around ten of each, where before it was around fifteen yes and fifteen no.
Maybe is off by default for that reason. If you are asking because you cannot decide, an answer that amounts to “still undecided” comes up a third of the time and leaves you where you started. It earns its place when the third answer is a real option: a turn in a game where a player is allowed to pass, or a plan that could happen today, not at all, or next week.
Pressing Ask again until it agrees with you
Asking again is a completely fresh draw, so nothing stops you from tapping until you see the answer you were hoping for. Each tap is still one in two. But if a no makes you reach for the button straight away, the picker has done something useful anyway: it showed you which way you were leaning before you admitted it. For a choice between two things you genuinely do not mind about, such as which film to watch first, take the first answer and move on.
Where the randomness comes from
This tool runs entirely in your browser, using its built in cryptographic random number generator. Nothing you type is sent to us, nothing is stored, and each result is thrown away the moment you leave the page. Numbers are drawn without the rounding bias that the usual shortcut introduces, so every outcome is as likely as every other.
Because nothing is recorded, a result here proves nothing to anyone afterwards. If you need a drawing that other people have to accept, for a prize, a campaign or an audit, use Winner Picker, which publishes each result on a permanent link with a signed record anyone can check.