Question

A population can be divided into two subgroups that occur with probabilities \(60 \%\) and \(40 \%\), respectively. An event \(A\) occurs \(30 \%\) of the time in the first subgroup and \(50 \%\) of the time in the second subgroup. What is the unconditional probability of the event \(A\), regardless of which subgroup it comes from?

Short answer

Answer: The unconditional probability of event A occurring, regardless of which subgroup it comes from, is 0.38 or 38%.

Step-by-step solution

Identify the given probabilities

Let's denote the two subgroups as events \(B_1\) and \(B_2\). The given probabilities are:$$ P(B_1) = 0.6 (60 \%) $$ $$ P(B_2) = 0.4 (40 \%) $$ $$ P(A | B_1) = 0.3 (30 \%) $$ $$ P(A | B_2) = 0.5 (50 \%) $$Here, \(P(A|B_1)\) represents the probability of event A occurring given that event \(B_1\) has occurred, and \(P(A|B_2)\) represents the probability of event A occurring given that event \(B_2\) has occurred.

Apply the Law of Total Probability

According to the Law of Total Probability, the unconditional probability of event A can be calculated as follows: $$ P(A) = P(A|B_1) \cdot P(B_1) + P(A|B_2) \cdot P(B_2) $$Substituting the given probabilities into the formula, we have: $$ P (A) = 0.3 \cdot 0.6 + 0.5 \cdot 0.4 $$

Calculate the unconditional probability of A

Now we can evaluate the expression for the unconditional probability: $$ P(A) = 0.18 + 0.20 $$ $$ P(A) = 0.38 $$The unconditional probability of event A occurring, regardless of which subgroup it comes from, is equal to \(0.38\) or \(38 \%\).

Key concepts

Conditional Probability

Conditional probability helps us to understand the likelihood of an event occurring, given that another event has already occurred. Let's break it down simply. When we say "given," we are talking about a condition that is already met. For example, if you only eat ice cream when it's sunny, the chance of you eating ice cream depends on it being sunny. That's conditional probability. It allows us to update our predictions based on new information or a specific situation. In mathematical terms, if we have two events, say Event A (eating ice cream) and Event B (sunny day), the conditional probability of Event A given Event B is denoted as \(P(A|B)\). This represents the probability of Event A happening provided Event B has indeed occurred.

• This concept is integral in everyday situations, where conditions or information influence the chance of events.
• It forms the foundation for more complex probability topics, like the law of total probability and Bayes' theorem.

Unconditional Probability

Unconditional probability, also known as marginal probability, evaluates the chance of an event happening without any conditions or additional information. It's like asking, "What's the chance of rain today?" without looking at the weather forecast. This form of probability ignores any prior circumstances, revealing the broader or overall likelihood of an event. Using our previous example, if we want to know the probability of eating ice cream without checking the weather, we're dealing with unconditional probability. In mathematical terms, for an event A, it's simply \(P(A)\), which reflects the chance of A occurring in the entire probability space.

• Unconditional probability is vital for general predictions or when detailed conditional information is unavailable.
• It's often calculated using available data or sometimes derived from conditional probabilities using rules like the law of total probability.

Probability Theory

Probability theory is an essential branch of mathematics that deals with quantifying uncertainty. It's like having a set of rules or guidelines to determine the likelihood of different outcomes. This theory builds the groundwork for reasoning in everyday decision-making and across various fields like statistics, finance, and science. Key components of probability theory include:

• Probability Space: The entire "universe" of possible outcomes for a given experiment or situation.
• Random Variables: Variables that can take different values based on the outcomes of chance.
• Events: These are specific outcomes or groups of outcomes we are interested in.
• Probability Measures: Functions that assign a likelihood to events within the probability space.

Probability isn't always intuitive, but probability theory provides frameworks and theorems, like the law of total probability, to calculate and understand it. By applying these principles, one can make informed predictions and decisions even in uncertain situations.