Question

A friend makes three pancakes for breakfast. One of the pancakes is burned on both sides, one is burned on only one side, and the other is not burned on either side. You are served one of the pancakes at random, and the side facing you is burned. What is the probability that the other side is burned? (Hint: Use conditional probability.)

Short answer

The probability that the other side of the pancake is also burned is 1/2.

Step-by-step solution

Identifying Possible Events

We start by identifying all the possible events. Two of the three pancakes have at least one side burned (the double-burned pancake and the single-burned pancake). Hence, the probability of getting a pancake with at least one side burned is 2/3.

Determine the Favorable Events

The next step is to determine the favorable event. In this case, the favorable event is getting a pancake that is burned on both sides. However, since we know the chosen pancake is burned on at least one side, we only consider those that meet the given condition, that is the double-burned pancake and the single-burned pancake.

Calculating Final Probability

Thus, the required probability is the ratio of the possibility of the desired outcome (double-burned pancake) to all possibilities given the observed condition (pancakes burned on at least one side). Among the pancakes that can be offered, one is burnt on both sides while the other one only on a single side. This gives us a probability of 1/2 that the other side of the pancake is also burnt.

Key concepts

Understanding Probability Theory

Probability theory is a branch of mathematics that deals with calculating the likelihood of various events. At its core, it attempts to quantify the uncertain by assigning numerical values to the chance of occurrence of different outcomes. An event represents a result or an outcome that might take place, and the probability of any event is usually expressed as a number between 0 and 1, inclusive. A probability of 0 means the event will not occur, while a probability of 1 indicates certainty. A probability of 0.5 is indicative of a situation that is as likely to occur as not.

In the pancake example from the textbook, the focus is on identifying the chance that a selected pancake has a burned other side when you already see one burned side. This scenario throws light on a critical aspect of probability theory: the analysis of events based on the given conditions or, as it's termed in the field, conditional probability.

The Process of Probability Calculation

Probability calculation involves mathematical techniques to determine the chances of various outcomes. The first step typically involves defining the sample space, which is the set of all possible outcomes. Next, you figure out which of those outcomes are favorable, or align with the event you're interested in. The probability of an event is then calculated by dividing the number of favorable outcomes by the total number of outcomes in the sample space.

In the case of the pancake example, we start by identifying that there are three pancakes with different conditions of being burned. Then, we narrow down to the specific scenario (condition) where the pancake is already known to be burned on one side. This influences our sample space and, consequently, our calculation, leading us onto the pathway of conditional probability. We end up with a simplified scenario where the probability calculation involves only the outcomes which adhere to our observed condition.

Navigating Event Outcome Analysis

Event outcome analysis is a key part of probability theory where the potential outcomes of an event are scrutinized. It requires a detailed approach in exploring the conditions surrounding an event to accurately predict the chances of different results. This involves not just counting possible outcomes but understanding how different conditions can shape the occurrence probabilities. It's akin to filtering results that are no longer possible and focusing on what remains given the new information.

Applying event outcome analysis to the pancake problem, one must consider only the relevant outcomes wherein at least one side is burned, as those are the only scenarios observable when you notice a burnt side facing up. It is an application of conditional probability: Given that one side is burned, we only analyze the outcomes that meet this condition. Through careful analysis, we are led to the conclusion that this limits our pool of outcomes to two possibilities, forming the basis for the subsequent probability calculation.