We can often change the odds in our favor. Here we have two decks with two cards each. To add up to five or more, we could draw the white three and the black two or the white three and the black three.
If we remove the white one from the white deck, then we maximize the probability that the goal is met since all combinations would then equal five or more.
Here's two different decks. To add up to five or more, we could draw the white two and the black three or the white three and the black three. So, we should remove either the white one or the black one to maximize our chances of meeting the goal. If we remove the white one, then our probability of meeting the goal is 2 out of four. If we remove the black one, then our probability of meeting the goal is 2 out of three. So, removing the black one maximizes the probability of meeting the goal.
These are the same two decks as before, but now our goal is to draw a white and black card that sum to exactly four. To sum to four, we could draw the white one and the black three, or the white three and the black one. In order to maximize the probability that we meet the goal, we can remove the white two.
This is the main idea. We're trying to eliminate cards that don't meet the goal without getting rid of pairs of cards that do. In our example, the goal was a sum of four. The white two was unhelpful because it couldn't make four with either the black one or the black three.
By removing it, we got rid of two losing combinations.
Let's look at a different scenario.
Here, there are two decks, each with three cards. Our goal is to draw a white card and a black card that sum to five.
The possible combinations are a white one and a black four, or a white two and a black three. Our starting probability of meeting the goal is 2 out of nine since there are two successful outcomes and 3 * 3 or nine total outcomes. The white three and the black one do not contribute to the goal. If we remove either one, we're left with a probability of two out of six possibilities.
Let's try one more. Here we have a deck of four white cards and a deck of three black cards. Our goal is to draw a pair of cards that sum to exactly six. The possible combinations are a white two with a black four, a white three with a black three, and a white four with a black two.
The white five is not part of any successful outcome. So removing it maximizes the probability the goal is met. The starting probability was three out of 12 possibilities or 1/4. Removing the white five results in three out of nine possibilities or 1/3, which is a higher probability. Ultimately, the best card to remove always depends on the specific numbers in play and the goal you're trying to achieve. By systematically analyzing how each removal changes the set of possible outcomes, we can find a choice that maximizes our probability of success.
