Welcome to Mini Week, a special week where I will be posting a new mini-article every weekday. Is this a good idea? Or a terrible one? I’m not quite certain. Depending on the feedback and whether writing five posts in a week proves fatal, it may become a one-time experiment or a recurring event. Let’s see how it goes!

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Picture this scenario: As you stroll through a distant land one day, you come across a plum tree. Excited, you pluck a plum from the tree. Before you can take a bite, a man nearby accuses you of stealing his plum.

“You took one of my plums,” he declares.

You try to return the plum, explaining that you did not intend to steal from him, but simply had a craving for plums and picked one.

“Enjoying plums and stealing one are not mutually exclusive,” he retorts. “It seems like you both wanted a plum and intended to steal one from me.”

“I guess so, but…” you begin.

“The penalty for stealing plums in our land is death,” he announces.

“Oh no,” you mutter.

He leads you to a tree stump, asks you to sit down, and pulls out three jelly beans – green, red, and blue. Placing them in a row on the stump, he explains, “Here’s the deal. Two of these jelly beans are poisonous, and eating either one will result in your death within 30 seconds. However, one jelly bean is safe and delicious. You must choose one jelly bean to eat. If you select the non-poisonous one, you’re free to go.”

“Cool,” you reply. It’s decision time – which jelly bean will you choose?

You opt for the green jelly bean.

Before you consume it, the man interjects, “Hold on. We have another tradition with prisoners. I will remove one jelly bean from the stump and put it back in my pocket. I’ll specifically remove a poisonous blue jelly bean because I know which colors are poisonous. The blue jelly beans are harmful.”

He removes the blue jelly bean and retains the red jelly bean on the stump, while the green jelly bean remains in your hand.

“It’s time to eat your jelly bean,” he instructs. “You have 10 seconds to decide whether to stick with the green one or switch to the red one.”

So, which jelly bean do you choose to eat? Does it make a difference?

Pause to contemplate your decision before proceeding.

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Here’s the surprising twist – your choice matters significantly. With just the information available in that moment, there is a distinct optimal choice: the red jelly bean.

You probably stuck with your initial decision to eat the green jelly bean because, with only two jelly beans remaining, a 50/50 chance seemed preferable, and loyalty to your original choice felt right. Correct?

Wrong. Very wrong.

The green jelly bean is twice as likely to be lethal as the red jelly bean. Here’s why:

Initially selecting the green jelly bean carried a 1/3 probability of it being safe and a 2/3 likelihood of being poisonous, with the safe jelly bean still on the stump. Removing the poisonous blue jelly bean from the stump did not alter the odds of the green jelly bean in your hand – it still had a 1/3 chance of being safe. However, this removal provided crucial information about the red jelly bean – if the safe one was originally on the stump, it must be the red jelly bean.

In essence, selecting a poisonous jelly bean (which occurs two-thirds of the time) and then switching post-removal guarantees your survival every time. If you initially picked the safe one (a one-third chance), switching would be fatal. Therefore, switching is the superior choice in two-thirds of the scenarios.

You could conduct a simple simulation: If you chose the green jelly bean 300 times, about 100 occasions would be safe to consume it, resulting in death if switching to red. Conversely, in the other 200 instances where the green jelly bean is poisonous, switching to red would save you every time.

An alternative perspective is to consider a scenario with 1,000 jelly beans. After selecting one (e.g., 267), and the removal of 998 poisonous beans, leaving jelly bean 749, the logical choice is to switch. Jelly bean 267 only holds a 1/1,000 chance of being safe, while 749 has persevered through the elimination process and is the survivor – significantly increasing the likelihood of it being the safe choice.

The same principle applies in the three jelly bean scenario. The isolated probability of the green jelly bean being safe (1/3) remains unchanged during the removal process, but the red jelly bean has effectively weathered the selection process and consolidates all potential safe choices. This is why switching to the red jelly bean is the recommended course of action.

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While this jelly bean dilemma is a fictional construct, it draws parallels to the renowned Monty Hall problem, popularized by Marilyn vos Savant in her “Ask Marilyn” column in 1990.

Monty Hall, host of the game show “Let’s Make a Deal,” presented a similar scenario:

Three doors – labeled 1, 2, and 3 – with a new car behind one and a goat behind each of the others.

After picking a door (e.g., Door 2), Monty, aware of the car’s location, reveals a goat behind another door (e.g., Door 3). He then offers the option to switch from the initial choice before unveiling a door.

Marilyn elucidated why switching to Door 1 was the correct move – as there was a 2/3 chance of initially selecting a goat, making the alternate door highly probable to conceal the car.

Despite Marilyn’s convincing reasoning, she faced backlash from over 10,000 individuals, including many PhDs, challenging her conclusion. The typical misinterpretation of this problem underscores the inclination to believe in a 50/50 probability for the final choices.

It’s crucial to note that the problem only functions under specific circumstances:

1) The host must always open a door not chosen by the contestant.

2) The host must be aware of the car’s location and reveal a goat behind the selected door.

3) The contestant must have the option to switch doors.

Any deviation from these conditions alters the outcome of the problem. For instance, with an unknowledgeable host randomly revealing a door, the odds shift to 50/50.

Another variant of this concept is the Three Prisoners problem, which can be explored further in the footnote for interested readers.

So, has the realization dawned on you? Are you on board, or are you still stuck in 50/50 territory?

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If you enjoyed this, you might like these other Blue Jay Blog posts:

What could you buy with $241 trillion? – A pizza the size of Niger is just one possibility.

7.3 billion people, one building – Quite the tight squeeze.

Graham’s Number: The biggest number ever – Truly mind-boggling.

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