How can probability be used in the decision tree process?

1 What is probability?

Probability is the measure of how likely an event or outcome is to occur, based on the available information and assumptions. It can be expressed as a number between 0 and 1, or as a percentage or fraction. For example, if you toss a fair coin, the probability of getting heads or tails is 0.5, or 50%, or 1/2. Probability can help you estimate the likelihood of different outcomes in a decision tree, and compare them based on their value and risk.

2 How to assign probabilities?

When assigning probabilities to the branches or nodes of a decision tree, there are various methods to choose from depending on the type and source of information you have. Empirical probability, for example, is based on observed data or historical records; if you know that 80% of customers buy product A and 20% buy product B, you can use this information in your decision tree. Subjective probability, on the other hand, is based on your personal judgment or opinion, or the opinion of experts or stakeholders. If you're launching a new product and don't have any data, you can assign probabilities based on market research, intuition, or customer feedback. Finally, theoretical probability relies on mathematical models or formulas that describe the behavior of a system or phenomenon; if you want to estimate the probability of completing a project on time, you can use a formula that takes into account the duration and variability of each task.

3 How to calculate expected value?

Expected value is the average value of an outcome, weighted by its probability, and can be used to compare different options in a decision tree. To calculate the expected value of an option, you need to multiply the value of each possible outcome by its probability and then add them up. For instance, if you have two options to invest your money, A and B, with respective outcomes and probabilities of $10,000 with 90% probability or $0 with 10% probability for option A; and $15,000 with 50% probability or $5,000 with 50% probability for option B; then the expected value of option A is ($10,000 x 0.9) + ($0 x 0.1) = $9,000; while the expected value of option B is ($15,000 x 0.5) + ($5,000 x 0.5) = $10,000. Consequently, option B has a higher expected value than option A.

4 How to account for risk?

Risk is the degree of uncertainty or variability in the outcomes of a decision, and it can affect how you prioritize and choose options. People with a risk-averse attitude prefer options with lower risk and lower expected value, while those who are risk-neutral only consider expected value, and those who are risk-seeking prefer options with higher risk and higher expected value. To account for risk in a decision tree, you can use sensitivity analysis to see how changing the probabilities or values of the outcomes affects the expected value and optimal decision. Additionally, you can apply a risk premium or discount by modifying the expected value based on your risk attitude. Lastly, you can use decision criteria such as maximin or minimax regret to select the option with the highest minimum outcome or lowest maximum regret.

5 How to use probability in practice?

To use probability in the decision tree process, you need to define your goal and alternatives, identify the uncertainties and outcomes, assign probabilities and values, calculate expected values and risks, and compare and choose according to your criteria and preferences. For example, a simple decision tree can help you decide whether to buy a lottery ticket or not. The goal is to maximize your net income. The alternatives are to buy or not buy the ticket. The uncertainties and outcomes are the chance of winning or losing the lottery, and the amount of money you gain or lose. The probabilities and values are based on the information given by the lottery. The expected values and risks are calculated using formulas, then compared in order to make a choice based on the expected value criterion.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?