Let's explore how to find probabilities for two card totals. First, let's figure out how many total outcomes are possible. Our first deck has three cards, 1, 2, and three. The second deck also has three cards, two, three, and four. To find all the possible pairs we can draw, we multiply the number of cards in each deck. So, three choices from the first deck times three choices from the second gives us nine total outcomes.
Now, let's set a goal. We want the sum of the two cards we draw to equal five.
Here are all the possible pairs we could draw. If we draw a one from the first deck, we need a four from the second to make five.
If we draw a two from the first deck, then we need a three from the second deck. And if we draw a three from the first, we need a two from the second.
That gives us three successful combinations. One and four, two and three, and three and two.
The probability of success is the number of successful outcomes divided by the total number of possible outcomes. We found three successful outcomes and nine total outcomes. So, the probability of drawing two cards that add up to five is three out of nine.
Visualizing all the possible outcomes is helpful for calculating probabilities.
Let's try this again with different decks. This time, the first deck has two cards, a two and a four. The second deck has four cards, 1, three, five, and six.
To find the total number of outcomes, we multiply the number of options in each group. That's two cards from the first deck times four cards from the second, which gives us eight total outcomes.
Now, our goal is to draw a white card that is less than a black card. Here are all the pairs we could get.
If we draw the two from the first deck, we can pair it with the three, five, or six from the second to meet the goal. If we draw four from the first deck, we could pair it with the five or six from the second.
Now, let's calculate the probability. We have five pairs that meet our goal out of a total of eight possibilities. So, the probability of drawing a white card that is less than a black card from these decks is five out of eight or 5/8s.
We found the probability by looking at all the possible outcomes and counting which ones met our goal.
