What Is a Tree Diagram?

Treemap, also known as dendrite. A tree diagram is a graphical representation of a data tree that organizes objects in a parent-child hierarchy. It is an expression of enumeration. The tree diagram is also a kind of graphics that junior high school students need to learn about probability problems.

Tree

In order to use a graph to show kinship, the taxonomic unit is placed on the top of a branch on the graph, and the relationship can be expressed according to the branch. It has two and three dimensions. in

Minimal tree diagram

Example 1. Randomly roll two cube cubes of uniform texture. The six sides of the dice are engraved with 1 to 6 points. What is the probability that the points on the upward side of the two dice are odd?

Analysis: The event in this question is to roll two dice. Looking at the points on the upward side, it can be determined that this event includes two links, the first dice and the second dice, so the tree diagram should draw two layers. The number of points on the upward side of the first dice may be one of six such as 1, 2, 3, 4, 5, 6, and so the first layer should draw 6 forks; look at the second layer, the second dice, The number of points on the upper side may be one of six, so the second layer should be connected to the six forks of the first layer, and each branch has six more forks. Draw a tree diagram to get a total of 6 × 6 cases, find out the cases where the points on the upward side of the two dice are odd, and then find the probability.

Solution: Draw a tree diagram, as shown in Figure 1.

It can be seen from the figure that there are 36 possibilities for the points on the upward side of the two dice, and the upward side points are all odd. , (3,1), (3,3), (3,5), (5,1) (5,3) (5,5) There are 9 cases, so the points on the side of the two dice are The odd probability (recorded as event A) is P (A) = 9/36 = 1/4

Comment: As can be seen from this example, as long as the tree diagram is drawn, it is easy to find the probability. The key to drawing a tree is to determine the number of layers, and the second is to determine the number of forks in each layer.

Example 2. Three red, white and green balls are contained in one pocket, and two red and white balls are contained in the other pocket (except the color). Now take a ball from each of the two pockets, and what is the probability that the two balls are the same color?

Analysis: This problem takes a small ball from each of the two pockets, which shows that there are two links in the event, and the tree-like picture has two layers. Since there are three small balls in one pocket, there should be three furcations on this floor. There are two small balls in the other pocket. There are two possibilities to get a ball. This layer may have two branches. The bifurcation of each layer should be drawn correctly. As for which one of the two pockets is put on and which is put down, you can draw freely without affecting the result.

Solution: Draw a tree diagram as shown in Figure 2 (Figure 3 is also correct):

It can be seen from the figure that there are 6 possibilities for the colors of the two small balls, and there are only 2 possibilities for the same color of the two small balls (red, red), (white, white), so two small balls The probability of the same color (recorded as event A) is P (A) = 2/6 = 1/3

Comment: The difference between this question and the previous question is that the number of balls in the two pockets is different, so the bifurcation of each layer should be determined according to the possible situation of this layer. This is the most difficult knowledge to draw in the tree diagram. try to figure out.