Let's look at how to figure out the chances of getting different totals when we add numbers from cards. We'll start with a single deck of four cards.
If we draw one card, is it more likely that we draw exactly a three or a number that is equal to or greater than three? The goal exactly three can only happen one way by drawing the three card. The goal equal to or greater than three can happen two ways. By drawing the three or by drawing the four. Since there are more ways to get a value of at least three, that outcome is more likely.
Let's look at a different problem. We'll draw one card from a white deck and one card from a black deck. Both decks have cards numbered 1 2 and three. Is the total sum more likely to be exactly five or at least five? A total of exactly five can happen in two ways. A white two and a black three or a white three and a black two. The goal at least five includes all the ways to get a five plus all the ways to get any total greater than five. The only total greater than five we can get is six by drawing a three from each deck. So the ways to get at least five are white two and black three, white three and black two, and white three and black three. That's three different ways. Since there are more ways to get a total of at least five, it's the more likely outcome. What if we compare two different exact totals? Which is more likely? A total of five or a total of six? There are two ways to get a total of five. a two and a three or a three and a two. There's only one way to get a total of six, a three and a three. Since there are two ways to make five versus only one way to make six, a total of five is more likely.
Here's an illustration of the possible ways we can get a total of five versus a total of six. Since there are two pairs that make a total of five and only one pair that makes a total of six, it's more likely to make a total of five.
Now, which is more likely, a total of two or a total of three? A total of two can only happen one way. A one from the white deck and a one from the black deck. A total of three can happen two ways. A white one and a black two or a white two and a black one. So, a total of three is more likely.
Now, let's compare the outcomes for a total of four versus a total of five. A total of four could have a 1 and a three, a two and a two, or a three and a one. That's three different ways to get a total of four. There are only two ways to get a total of five. So, a total of four is more likely than a total of five. Not every two card total is equally likely. The chance of getting a certain total depends on how many different pairs of cards add up to that number.
