Changing the cards in a deck can change the probability that you'll get the result you want. This is sometimes called stacking the deck. Let's start with a simple deck of four cards numbered 1 through 4. The goal is to draw a card with an odd number. Right now, there are two odd cards, the one and the three out of four total cards.
The probability is two fourths or 1/2.
Let's remove one card to make drawing an odd card more likely. We could remove an odd card like the three, but then we'd only have one odd card left in the three card deck, a one in three chance, which is worse. The better strategy is to remove a card that doesn't meet our goal, like an even card. Let's remove the two. Now, we still have two odd cards, but the deck only has three cards in total. Our probability of drawing an odd card has increased from 2/4s to 2/3.
Let's look at another example. Here are two decks of three cards. The goal is to draw a white card with a value of two or more and a black card with a star. To make the goal more likely, we can remove one of the cards that doesn't meet the conditions. There are three cards we could remove. The white one, the black triangle, or the black square. Now, let's remove a card to maximize the probability of the goal being met. The goal is to draw a white card with a circle and a black card with a three.
There are two cards that do not meet the goal. A white triangle card and a black two. Let's see what happens if we remove the white triangle card. We'd then have two successful outcomes. One of the white circle cards with the black three out of four possible outcomes. If we remove the black two card, then we'd have two successful outcomes. One of the white circle cards and the black three out of three possible outcomes. To maximize our chances, we should remove the black two since 2/3 is greater than two/4s.
When we remove a card, we have to consider how it affects both the number of successful outcomes and the total number of outcomes. Removing the black two resulted in lowering the number of total outcomes from six to three. While removing the white triangle only lowers the number of total outcomes from six to four.
Let's apply this idea to a different scenario. The goal is to draw a white card with a value less than or equal to three and a black card equal to three.
The white four and black one cards don't meet the goal. However, removing the white four would result in six out of nine successful outcomes, while removing the black one would result in six out of eight possible outcomes. To maximize the goal, remove the black one card.
Modifying the smaller deck of black cards had a bigger proportional effect on the probability than modifying the larger deck of white cards.
