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Conditional probability is a fundamental concept in probability theory that describes the likelihood of an event occurring given that another event has already occurred. This notion can transform an uncertain scenario into one with more information, thereby affecting the outcome's probability. It is formally defined as the probability of an event A given the occurrence of another event B and is mathematically represented by the notation P(A|B).

In the context of the stock price movement exercise, conditional probability is crucial for understanding part (c) of the problem. After observing that the stock's price increased by 1 unit after 3 days, we're asked to calculate the likelihood that the price went up on the first day. Here, we're dealing with a conditional probability because the occurrence of the event 'increase by 1 unit after 3 days' provides us with additional information to reassess the probability of the first day's movement.

Using Bayes' theorem, a powerful formula in probability, we can determine such probabilities by relating the likelihood of the 'condition' (the increase after 3 days) to the likelihood of the 'target event' we are interested in (the stock going up on the first day). It can be particularly helpful when dealing with a series of independent events, where each event does not influence the occurrence of others.

In probability theory, events are independent if the occurrence of one event does not affect the likelihood of the other events. This means that the occurrence of one event gives no information about whether the other event will occur; the probabilities are unaffected.

For example, consider the toss of a fair coin: whether it lands heads or tails on the first toss has no influence on what it will land on the second toss. The probability of getting heads on the first toss is the same regardless of what happened in previous tosses. Independent events are a key assumption in many statistical models as they greatly simplify the probability calculations.

In the exercise we are examining, the daily stock price movements are assumed to be independent events. This independence allows us to easily compute the probability of the stock returning to its original price after 2 days (part a) or of the stock increasing by 1 unit after 3 days (part b) by simply multiplying the individual probabilities of each day's movement.

A stock price movement model is a mathematical framework used to describe and predict the changes in stock prices over time. In the simplified model given in our exercise, the stock price on each day moves either up or down by 1 unit. The probability of the price going up on any given day is represented by 'p', while the probability of it going down is '1-p' due to the law of total probability which states that all possible outcomes must sum up to 1.

The given model assumes that the movements on different days are independent events, which means that the movement on any given day does not influence the movement on any other day. This simplifies the computation of probabilities over multiple days. In practice, such models can become very complex, often considering more variables and relying on different theories of financial economics, like the random walk hypothesis or the efficient-market hypothesis.

Understanding such models is crucial for aspiring financiers and statisticians as they provide basic foundational knowledge towards comprehending more advanced financial instruments and their respective pricing models.