To understand probability, we first need to know how to count all the possible results or outcomes. An outcome is one specific way something can happen. Let's see how to count them. Starting with a single deck of cards. Here's a deck with three cards. A one with a star, a two with a square, and a three with a square. We want a goal that can be met in exactly two ways. If the goal is drawing a three, there's only one card that works. But if the goal is drawing a square, both the two and three cards fit. That gives us two ways.
Now, let's draw one card each from two different decks. There is only one way to draw a three from the first deck and a square from the second. Since the first deck only has one three and the second only has one square. If we wanted to draw a square from the first deck and a two from the second, there are two different ways because the first deck has two squares and the second deck has one two.
Let's figure out how many ways we can meet these goals. To draw a square from the first deck and a one from the second, we have four ways because there are two squares in the first deck and two ones in the second. There are three cards less than or equal to three in the first deck and two one cards in the second deck. So there would be six ways to draw a card less than or equal to three from the first deck and a one from the second deck.
Each way that cards can be drawn is an outcome. When you combine choices from separate groups, you find the total number of outcomes by multiplying.
Let's apply this. Our goal is to draw a card with a square from the first deck and a card that's less than or equal to two from the second deck. How many outcomes meet this goal? First, let's count the matching cards in the first deck. There are two cards with a square.
In the second deck, all three cards are less than or equal to two. To get the total, we multiply the options from each deck. 2 * 3 equals six possible outcomes that meet the goal.
So far, we've counted outcomes for a specific goal, but we can also use multiplication to find the total number of all possible outcomes. To find how many different pairs we can draw from two decks, we simply multiply the total number of cards in each deck. Here, both decks have three cards. So, the total number of possible outcomes is 3 * 3, which equals 9. There are nine different pairs you can possibly make.
Here, one deck has two cards and the second deck has four cards. To find the total number of outcomes when drawing one card from each, we can multiply the number of cards. That's two from the first deck time four from the second, which gives us a total of eight possible outcomes.
To find the total number of outcomes, we multiply the number of options in each group. To find the number of outcomes for a specific goal, we multiply the number of choices we have in each group that meet the goal.
