Tree of Possible Outcomes

This material is suitable for children of any school age. It helps develop auditory perception through solving mathematical tasks and trains auditory memory.

We focus on logical tasks aimed at developing reasoning skills and the ability to solve non-standard problems.

A tree of possible outcomes is a graph that represents a multi-step decision-making process.

Task #1

John takes a walk from point A along the park paths. At each fork in the road, he randomly chooses the next path without turning back. The path diagram is shown in the illustration. Find the probability that John will reach point G.

Solution: Next to each edge, we write the probability that John will take the corresponding path. The choice at each fork is random, so the probability is equally divided among all possibilities. Each route from the starting point A to any of the final points is an elementary event in this experiment. The events here are not equally probable. The probability of each elementary event can be found using the multiplication rule. This event consists of John taking the route ABG. The probability is found by multiplying the probabilities along edges AB and BG. 0.5 · 0.25 = 0.125.

A tree is a graph without cycles, meaning you can’t travel along multiple edges and return to the starting point. Constructing a tree is a simple way to solve non-standard problems in probability theory.

Task #2

To list all three-digit numbers consisting of the digits 1 and 2, you can use a graph.

By building this sequence, we find all the three-digit numbers. The total number of options always equals the number of points in the last row.

Task #3

In the Magic Land, there are two types of weather: good and excellent, and once the weather is set in the morning, it remains unchanged throughout the day. It is known that with a probability of 0.9, tomorrow’s weather will be the same as today’s. On April 5, the weather in the Magic Land is good. Find the probability that the weather on April 8 will be excellent.

Solution: For the weather on April 6, 7, and 8, there are four possibilities: GGE, GEE, EGE, EEE (here G — good, E — excellent weather). We will find the probabilities for each weather outcome:

• P(GGE) = 0.9·0.9·0.1 = 0.081
• P(GEE) = 0.9·0.1·0.9 = 0.081
• P(EGE) = 0.1·0.1·0.1 = 0.001
• P(EEE) = 0.1·0.9·0.9 = 0.081

These events are mutually exclusive, so the total probability is the sum of the probabilities of these events: P(GGE) + P(GEE) + P(EGE) + P(EEE) = 0.081 + 0.081 + 0.001 + 0.081 = 0.244.

Answer: 0.244.