Here’s your Micro Softy for Monday! It’s a bit like the last one (see the solution below). It comes down to a question of when and how to stop making new choices.

Making a deal

Let’s Make a Deal was a television game show, first hosted by Monty Hall (1921–2017) in 1963. The basic idea is that there are three doors and a contestant’s job is to barter with Monty for the most valuable prize behind the doors.

The Monty Hall problem, as it came to be called, was loosely based on the quiz show. It was popularized by Marilyn vos Savant in her Parade Magazine column, “Ask Marilyn,” in 1990. A reader wrote to ask:

”Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, ‘Do you want to pick door No. 2?’ Is it to your advantage to switch your choice of doors?”

That’s our new Micro Softy.  What’s the answer?

Solution to Micro Softy 9:   To Flip or Not to Flip

As you may recall, in this problem, there are three cards turned face down. Each has a dollar amount written on it. The amount can be anything from a penny to millions of dollars. The first card, chosen at random, is turned over. You have the choice of keeping the amount on the card or ripping it up and turning over a second card. You look. Again, you can keep the amount on the second card or rip it up and turn over the last card.  The last card you turn over is the one you keep.

There’s a catch. You can only walk away the amount shown on a card if it’s the largest of the dollar amounts on all three cards. Otherwise, you win nothing.

Here’s the winning strategy that gives you a 50-50 chance of walking away with the money. Flip over the first card and reject it. Flip over the second card. If the amount is less than the first card, you know it is not the largest. So flip over and keep the third card’s amount. If, though, the second card is greater than the first, claim the amount on the second card.

Below is a table of all six possibilities of card flips. Card 3 has the greatest amount of money and Card 1 the least. The ordering 2-1-3 means Card 2 is chosen first, followed by Card 1. Each ordering is equally probable. “W” means win and “L” lose.

For example, consider the ordering 1-3-2. You first flip over Card 1 which has the least amount of money on it. You rip it up and turn over Card 3 that has the most money written on it. Since Card 3 is bigger that Card 1, you stop flipping and claim the amount on Card 3. You win and walk away with the money.

Each of the entries in the table can be analyzed in the same way.

• 123      L
• 132      W
• 213      W
• 231      W
• 312      L
• 321      L

Overall, there are 3 winning combinations and three losing ones. Since all are equally probable, the chance of walking away with money is one half (50-50).

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Here are the Micro Softies from earlier Mondays, to date. The answer to one Monday’s puzzle is always given the next Monday:

Monday Micro Softy 1: Microsoft: What did you need to work there in the early days? They asked questions that were not about the details of computer technology. The questions made you think. Hard. For example, they would ask questions like, why are manhole covers round? Also, here’s the Round Trip puzzle.

Monday Micro Softy 2: The Dead Presidents’ Club The answer to Monday Micro Softy 1: The Round Trip puzzle is here too. Today’s puzzle: How would a girl who knew nothing of American history immediately know the name of at least one of three early Presidents who died on July 4?

Monday Micro Softy 3: The Wolverton Mountain Puzzle Here’s the answer to Dead President’s Club as well — and smart STEM people often DON’T get that one right. Today’s brain teaser is in honor of Claude King, Clifton Clowers, and Wolverton Mountain. It’s not high tech but it will surely test your thinking abilities.

Monday Micro Softy 4: Claude King bests Clifton Clowers Clowers offers Claude two slips of paper to choose from on a blind choice: marriage or death… Claude’s sweetheart Chloe warns him that, in reality, both slips say “death.” He says never mind. So how did he escape death and marry her?

Monday Micro Softy 5: The puzzle of Claude and Chloe’s two kids Puzzle: We learn that one of Claude and Chloe’s two children is a boy. With a 50–50 ratio, what chance is there that the other child is a boy too? Also, here’s the answer to the puzzle of how, twelve years earlier, Claude escaped the trap Clifton Clowers set for him, so he could marry Chloe.

Monday Micro Softy 6: Bad Adding? It looked to Claude like young Clay’s numbers didn’t add up but the boy seemed confident. What did he know that his father at first didn’t?And here’s the answer to the probability question: What are the chances that Claude and Chloe’s younger child is also a boy?

Monday Micro Softy 7: Who’s the Champ? A single number does not always determine which player is better. And here’s the answer to why young Clay’s arithmetic didn’t make sense at first to Claude — but then Claude realized that the boy was right.

Monday Micro Softy 8: Who’s the better barber? There are only two barbers to choose from and the choice may not be as easy as it looks. Also, here’s the answer to Who’s the Champ?
Micro Softy 7 is an illustration of Simpson’s Paradox, which says that averaging averages doesn’t always work.

Monday Micro Softy 9: To Flip or Not to Flip? Probability theory can sometimes help with seemingly impossible questions. But how? About the answer to Micro Softy #8, ask yourself a question, “Who shaves the barber?”