**Math, Speculation and Intuition**

This is an important subject whenever faced with a news story or ad that tries to entice or scare you based on the chance you estimate events to be subject to.

Five decks of cards are arranged into two giant decks face down such that one is 70% diamonds and hearts while the other is 70% clubs and spades. You select one decks and draw 12 cards at random and find that 8 of the cards are red and 4 are black.

Question: what is the approximate likelihood that the deck you drew from is the deck that has mostly red cards?

The point is not to struggle with whatever calculation you can to get as close to a precise answer as possible, but to be aware of your thought processes whenever implicitly called upon to guess likelihood.

To really understand and flex your mind, you might just guess, and try to notice what it is that leads to your hunches about numbers. Was it a prior good or bad experience where there was uncertainty? How was it affected by the outcome and how surprising was it? The randomization of numbers does produce some surprising results regardless of whether there is any truth to “supernatural” influences, but that’s beyond the scope here.

Or maybe — especially if you have some skill with calculating odds — you almost unconsciously searched your history for experience with similar calculations and tried to see what kind of rough answer you could give with the time and energy you had. Below I’ll show the needed calculation so you can compare it with the process you did — or didn’t feel up to yet, but might now that you know the answer is coming. Actually it’s not nearly as complex as you might think — *if you frame what is really being asked*.

You might, rightly or wrongly, have determined that math is not a strength for you, but that doesn’t keep you from seeing that — especially without a spreadsheet program — it would be a major task to calculate the likelihood of all possibilities (for example, drawing 7 cards of the same color as those predominantly in the deck and 5 of the other color, instead of 8 and 4 — or 6 and 6, or 5 and 7, and so on — but if you pay close attention, you are not being asked to do anything of the kind — let alone calculate the *order* in which they are drawn). More about order and spreadsheets after dispensing with the word problem.

You are only being asked about the* relative* likelihood of only *two *outcomes: