Daily Math Puzzle: 2026-10-20
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2026-10-20
At the "Prize Booth Shuffle" game, there are three identical, sealed envelopes. One contains a "WIN" token, the other two are empty. You pick one envelope. Before you open it, the booth attendant adds a fourth identical, empty, sealed envelope to the *two remaining envelopes* on the table (not the one you picked). These three envelopes on the table are then shuffled. The attendant then offers you the choice: either keep your original envelope or swap it for *one* of the three shuffled envelopes on the table. What should you do to maximize your probability of winning?
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Solution
Stick with your original choice. — Let's analyze the probabilities:
**Scenario 1: You stick with your original choice.**
Initially, there are 3 envelopes, one with WIN. Your probability of having picked the WIN envelope is 1/3. If you stick with your original choice, this remains your probability of winning.
P(Win | Stick) = 1/3.
**Scenario 2: You swap your envelope for one of the three shuffled envelopes on the table.**
We need to consider two possibilities for your initial pick:
* **Case A: Your original pick was the WIN envelope.** (Probability = 1/3)
If you picked the WIN envelope, the two envelopes remaining on the table were both empty. The attendant then adds a fourth, empty envelope to these two. So now, the three envelopes on the table are *all empty*. If you swap your WIN envelope for any of these three, you are guaranteed to pick an empty envelope.
In this case, P(Win | original pick was WIN, Swap) = 0.
* **Case B: Your original pick was an EMPTY envelope.** (Probability = 2/3)
If you picked an EMPTY envelope, then the two envelopes remaining on the table contained one WIN and one EMPTY. The attendant then adds a fourth, empty envelope to these two. So now, the three envelopes on the table consist of one WIN and two EMPTY envelopes. If you swap and pick one of these three, your probability of picking the WIN envelope is 1/3.
In this case, P(Win | original pick was EMPTY, Swap) = 1/3.
Now, we combine these probabilities to find the overall probability of winning if you swap:
P(Win | Swap) = P(Win | original pick was WIN, Swap) * P(original pick was WIN) + P(Win | original pick was EMPTY, Swap) * P(original pick was EMPTY)
P(Win | Swap) = (0 * 1/3) + (1/3 * 2/3)
P(Win | Swap) = 0 + 2/9 = 2/9.
**Conclusion:**
P(Win | Stick) = 1/3 = 3/9.
P(Win | Swap) = 2/9.
Since 3/9 > 2/9, sticking with your original choice (1/3 chance of winning) gives you a higher probability of winning than swapping (2/9 chance of winning).
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