Day 1 - General Probability

Probability is the foundation of solving brainteasers involving randomness and uncertainty. Today, we’ll focus on core principles that will allow you to calculate probabilities and solve a wide range of problems with confidence. By understanding how probabilities are calculated, combined, and applied, you'll be equipped to handle many types of interview challenges.

Basic Probability

Probability measures the likelihood of an event occurring. The basic formula is:

Probability = (Number of Favorable Outcomes) / (Number of All Possible Outcomes)

For example, if you roll a standard 6-sided die, the probability of rolling a 3 is:

P(rolling a 3) = 1 / 6

This is because there is 1 favorable outcome (rolling a 3) and 6 possible outcomes in total (rolling any of the numbers 1 through 6).

Combining Events

In many problems, you'll need to combine probabilities of multiple events. It's crucial to know when to use P(A and B) vs P(A or B), as they represent very different concepts:

• P(A and B): This is the probability that both events A and B occur simultaneously. For example, rolling a 4 on the first die and a 5 on the second die are independent events, so:

  P(A and B) = P(A) * P(B)

  If P(A) = 1/6 and P(B) = 1/6, then:

  P(A and B) = 1/6 * 1/6 = 1/36

• P(A or B): This is the probability that either event A or event B occurs, or both. Here, you're exploring all possible ways that one or both events could happen. For example, if rolling a 4 or a 5 on a single die:

  P(A or B) = P(A) + P(B) - P(A and B)

  Since it's impossible to roll both a 4 and a 5 on one die, P(A and B) = 0. Thus:

  P(A or B) = 1/6 + 1/6 = 1/3

A common mistake is confusing these two concepts. Use P(A and B) when you're considering events that must happen at the same time, and P(A or B) when you're summing up all possible ways in which one event or the other could happen.

Law of Total Probability

The Law of Total Probability helps you find the chance of something happening by adding up all the different ways it could happen. For example, imagine you take the bus or the train to work. If it rains, you're more likely to take the bus. If it's sunny, you prefer the train. To find the total chance you take the bus, you look at both cases—rain and sun—and add: (chance of rain × chance of bus if rain) + (chance of sun × chance of bus if sun).

The general formula is:

P(A) = P(A|B₁)·P(B₁) + P(A|B₂)·P(B₂) + ...

Heuristics and Tricks for Solving Probability Problems

• Count Total Outcomes: Always start by identifying the total number of possible outcomes. For example, rolling two dice has $6 \times 6 = 36$ total outcomes because each die can land on any of 6 faces.
• Break Complex Problems into Simpler Steps: If a problem feels overwhelming, break it into smaller parts. For example, if you're finding the maximum of two dice rolls, first calculate how many ways each possible maximum value (1 through 6) can occur.
• Look for Symmetry: Many problems involve symmetry that simplifies calculations. For example, when rolling two dice, the outcome of (4, 5) is equivalent to (5, 4) in terms of probabilities. Use this symmetry to reduce redundant calculations.
• Understand the Key Condition for Success: In some problems, the "success" condition depends on geometric or positional relationships. For instance, if randomly placing points on a circle, success might depend on whether the circle's center lies within a certain region. Identify these conditions early to guide your calculations.
• Use Disjoint Events: If events cannot happen at the same time (disjoint), you can sum their probabilities. For example, the probability of rolling a 4 or a 5 on a single die is:

  P(4 or 5) = P(4) + P(5) = 1/6 + 1/6 = 1/3