Beat fifty with one redraw

Question

You draw a number uniformly at random from $1$ to $100$ . You win if your final number is greater than $50$ . After seeing your first draw you may keep it, or redraw once (uniform $1$ to $100$ ) and keep the new number.

What is your optimal rule, and what is your probability of winning?

Accepted Answer

Here the target is fixed at 50, so the decision is simple: if your first draw already clears 50 you have won, so keep it; if it does not, keeping guarantees a loss, so you must redraw. Keep if the first draw is above 50, otherwise redraw. Now the win probability. With probability 	frac12 the first draw already beats 50 and you win outright. With probability 	frac12 it does not, and you redraw, winning that redraw with probability 	frac12: P(	ext{win}) = \frac12 + \frac12 \cdot \frac12 = \frac12 + \frac14 = \frac34 = 0.75. The logic is the general keep-or-swap rule in its cleanest form: ke…

Beat fifty with one redraw

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