You are presented with two identical envelopes with unknown sums of money. One envelope contains a greater amount of money, however, you do not know which envelope and what amount this is.
You open the left envelope and count the money to be ₹4392. Should you risk swapping to the other envelope or keep this one?
This scenario was popularised by famous statistician, David Blackwell. He managed to work out a mindbogglingly unique solution to the dilemma where your odds of picking the larger sum of money is not 50-50.
The proposed solution is:
Pick a random number, any number. If this number is greater than 4392, then open the other envelope. If this number is smaller than 4392, keep this envelope.
Note : 4392 is not a constant value. It is the sum of money used by me in this thought experiment. It should be replaced with the money you actually in the envelope you opened.
Surprisingly at first, the simple act of picking a random number allows you to predict the envelope with more money with higher certainty than 50%. How?
Understanding the Mathematics
To understand why a seemingly unrelated random number improves our odds, let us draw a diagram. Let the amount in our letter be ₹a while the amount in the other letter be ₹b. Now, we must consider two cases : the other letter has more money, or the other letter has the less money. Both cases have a 50% chance of occuring.
Case 1 : Less money in other letter (b < a)
Case 2 : More money in other letter (a < b)
Blue section : Since b is some value away from a, there is some non-zero probability (P) that our random number we pick is between them. In other words, there is a chance, however small it may be, that the random number we pick lands between the values of the envelopes. If this does happen, then according to our strategy (Refer back to recall your memory) we will not switch in Case 1, but switch in Case 2. Hence, winning the greater amount of money both times.
Yellow section : This situation occurs when the random number we choose is less than both a and b. If this happens, according to our strategy, we will NOT switch in either case. Not switching is good for Case 1 but bad for Case 2. Since the probability of either case is 50/50, our number being in the yellow section makes us win 50% of the time.
Orange section : This situation occurs when the random number we choose is more than both a and b. If this happens, according to our strategy, we WILL switch in either case. Switching is bad for Case 1 but good for Case 2. Since the probability of either case is 50/50, our number being in the orange section makes us win 50% of the time.
To recap (Note : Refer to diagram simultaneously for more clarity):
Probability of winning = 100% if Blue, 50% if Yellow, 50% if Orange
= (100% * P) + (50% if Yellow or Orange)
= (100% * P) + (50% * (1 - P))
= P + 0.5 - 0.5P
= 0.5 + 0.5P > 0.5
As we see, the probability we win is (0.5 + 0.5P) and since P is some non-zero positive number, we have shown our probability is greater than 0.5!
Note : We are performing our calculations considering you could get a negative amount of money too. So, if you opened an envelope with just ₹2, then the other envelope would be just as likely to have a negative million rupees as having a positive million.
Proposal to Predict Tosses
Assign the right envelope heads and the left envelope tails. Have someone place the greater amount of money in the envelope that had the "correct side". Remove the blindfold and perform the Blackwell's Bet without physically opening the second envelope. You can now with a certainty of more than 50% predict the toss of the coin.
At face value, this proposal does indeed predict a toss with greater than 50% certainty after it has happened. The key problem is the "after it has happened" part. For the proposal to work, the knowledge of the coin toss must be known by a friend who places the money appropriately in the envelope acting as the major caveat of such a plan.
Expanding the applications
Nonetheless, Blackwell's Bet is a critical example of how a random event like picking a number can entangle two arbitrary events. The use of "entangle" here is of specific choice since it is borrowed terminology from quantum mechanics. (Note : In quantum mechanics, "entanglement" basically refers to how two particles, however far or close, are linked in a manner where measuring the "spin" of one particle reveals the "spin" of another.) This may be a new concept to grasp for most, hence I will try giving some examples to make it more palatable.
Select two random events that can have a "success/failure" outcome. For instance, Event 1 is the probability of a resident in Mumbai speaking Marathi. Event 2 is the probability of a resident in California being taller than 5'5". As you may notice, these events are drastically unrelated and should have no effect on each other. Surprisingly, one can figure out which probability is larger by using the following steps:
Flip a coin to determine which event you will choose to observe. Heads = Event 1; Tails = Event 2.
Let's say the coin lands heads. Now, you go out in Mumbai and pick an arbitrary person off the street to ask whether they speak Marathi.
If they do, then you can predict with more than 50% certainty that the Probability of Event 1 will be greater than Probability of Event 2
If they don't, then you can predict with more than 50% certainty that the Probability of Event 1 will be smaller than Probability of Event 2
Similar to how the picking of a random number in the Blackwell's Bet "entangled" the envelopes and gave you a chance to predict which is greater, the coin toss here "entangles" Event 1 and Event 2 to give you a chance to predict which is greater.
Conclusion
This Bet is just one of the many counterintuitive yet awe-inspiring results of mathematics. The expansion of the Bet into the field of "quantum entanglement" is weakly pursued and may even be an opportunity for a reader like you to explore their connections and make math and physics just that much more connected.
Bibliography and Suggested Reading
Ross, Greg. “Blackwell’s Bet.” Futility Closet, 28 June 2016, www.futilitycloset.com/2016/06/28/blackwells-bet/.
Stein, James D. Fate of Schrodinger’s Cat, The: Using Math and Computers to Explore the Counterintuitive. World Scientific, 2020. Chapter 3.
Stein, James, and Leonard Wapner. A Coin-Tossing Conundrum. arxiv.org/pdf/1710.01298.pdf.
Stein, Jim, guest. "Surprisingly Better than 50-50." The Art of Mathematics, Spotify app, Dec. 2022.
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