Question

Assume that \(P\left(A_{1} \cap A_{2} \cap A_{3}\right)>0\). Prove that $$ P\left(A_{1} \cap A_{2} \cap A_{3} \cap A_{4}\right)=P\left(A_{1}\right) P\left(A_{2} \mid A_{1}\right) P\left(A_{3} \mid A_{1} \cap A_{2}\right) P\left(A_{4} \mid A_{1} \cap A_{2} \cap A_{3}\right) . $$

Short answer

The given equation is proved by expressing each term on the right-hand side in terms of the intersection probabilities and simplifying, eventually obtaining \(P(A_1 \cap A_2 \cap A_3 \cap A_4)\) on both sides.

Step-by-step solution

Define the Required Probability Terms and Conditional Probability

The first step is to define the probability of an event and conditional probability. The probability of an event \(A\), denoted \(P(A)\), is a measure of the likelihood that the event will occur. If we are considering two events, the probability of their intersection, denoted \(P(A \cap B)\), is the probability that both events occur. The conditional probability of \(A\) given \(B\), denoted \(P(A|B)\), is the likelihood of event \(A\) occurring given that \(B\) has already occurred. It is calculated as \(P(A|B) = P(A \cap B) / P(B)\) (provided that \(P(B) > 0\)).

Apply the Conditional Probability Formula to Each Term

We now break down the right-hand side of the equation using the definition of conditional probability. Start with \(P(A_1)\): this remains the same as there are no conditioning events. Then, for \(P(A_2 | A_1)\), \(P(A_3 | A_1 \cap A_2)\), and \(P(A_4 | A_1 \cap A_2 \cap A_3)\), express each in terms of intersection probabilities. This will give us: \[ P(A_2 | A_1) = P(A_1 \cap A_2) / P(A_1) \] \[ P(A_3 | A_1 \cap A_2) = P(A_1 \cap A_2 \cap A_3) / P(A_1 \cap A_2) \] \[ P(A_4 | A_1 \cap A_2 \cap A_3) = P(A_1 \cap A_2 \cap A_3 \cap A_4) / P(A_1 \cap A_2 \cap A_3) \]

Compute the Required Probability

Now, multiply all the terms calculated in the previous step: \[ P(A_1) * P(A_1 \cap A_2) / P(A_1) * P(A_1 \cap A_2 \cap A_3) / P(A_1 \cap A_2) * P(A_1 \cap A_2 \cap A_3 \cap A_4) / P(A_1 \cap A_2 \cap A_3)\]. Here, you can observe that \(P(A_1)\) in the denominator cancels with \(P(A_1)\) in the numerator in the second part of the expression. Similarly, \(P(A_1 \cap A_2)\) and \(P(A_1 \cap A_2 \cap A_3)\) in the denominator cancel with their respective terms in the numerator in the third and fourth parts of the expression, respectively. Thus, you're left with \(P(A_1 \cap A_2 \cap A_3 \cap A_4)\) which was to be proved.

Key concepts

Probability of an Event

Understanding the probability of an event is essential in studying randomness and uncertainty. The probability of an event, denoted as \(P(A)\), is a numerical measure ranging from 0 to 1, where 0 indicates impossibility and 1 indicates certainty.

Intuitively, the probability of an event can be thought of as the long-run relative frequency of the event occurring during repeated trials of an experiment. When flipping a fair coin, for example, the probability of landing on heads, \(P(\text{Heads})\), is \(\frac{1}{2}\) because there are two equally likely outcomes and one of them is heads.

It's critical for students to recognize that the probability must be grounded in the context of the specific event and the total number of possible outcomes. This will help in both theorizing about and calculating probabilities accurately.

Intersection of Events

The intersection of events is a foundational concept in probability that deals with the occurrence of two or more events simultaneously. If you have two events, \(A\) and \(B\), their intersection, symbolized as \(A \cap B\), represents the set of outcomes that are common to both \(A\) and \(B\).

To illustrate, if \(A\) is the event of selecting a red card from a standard deck of playing cards and \(B\) is the event of choosing a king, then \(A \cap B\) would represent selecting the king of hearts or the king of diamonds - the overlap between red cards and kings.

When events are independent, meaning the outcome of one does not affect the other, the probability of their intersection is the product of their individual probabilities \(P(A) \times P(B)\). However, when events are not independent, this relationship does not hold, and conditional probabilities are used instead.

Probability Formulas

Probability formulas are mathematical expressions that quantify the chance of events occurring. There are several key formulas that students must become familiar with:

• For independent events \(A\) and \(B\), the probability of their intersection is given by \(P(A \cap B) = P(A) \times P(B)\).
• The Conditional Probability formula, \(P(A|B) = \frac{P(A \cap B)}{P(B)}\), calculates the probability of event \(A\) occurring given that event \(B\) has already occurred, provided \(P(B) > 0\).
• The Law of Total Probability and Bayes' Theorem are also important tools for handling complex problems involving conditional probabilities.

Grasping these formulas not only helps in solving textbook exercises but also enables students to tackle real-world problems where uncertainty plays a significant role. Conceptual understanding coupled with practice in applying these formulas fortifies one's command over the subject matter.