For example: when trying to find the probability that entering a bank with 3 bank clerks at random will take less than a minute given average times, we use the exponential distribution with $\lambda$.
How exactly is that $\lambda$ found or what is it based on when it comes to real world scenarios?
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This is a great question. First of all, there are infinitely many probability measures given a measurable space $(\Omega,\mathcal{F})$, however, like @Eff said, we usually concern ourselves with an "empirical" or "physical" measure $\mathbb{P}:\mathcal{F} \to [0,1]$ which is constructed from empirical data.
To see how two different probability measures might be used in the real world, let's say we are a market-maker in a fairly liquid options market. Our goal is to earn a profit by buying a given derivative at one price and then to quickly unload that same derivative at a higher price as quickly as possible. The whole point of this trading method is to ideally have zero exposure to the directional risk (i.e. the change in the price of the underlying asset) of the option and to have our profit derived completely from the bid-ask spread. To minimize this exposure, we take the empirical probability measure $\mathbb{P}$ and use the Radon-Nikodym theorem to construct an equivalent measure $\mathbb{Q}$ so that the payoff of a certain option, given by a random variable $H_{t}$ can be discounted between a given moment in time (i.e. $t=0$ and a time $T$ by a discount factor $D(0,T)$ via the relationship $$ H_{0}=D(0,T)\mathbb{E}_{\mathbb{Q}}[H_{T}] $$ and since $\mathbb{P}$ and $\mathbb{Q}$ are equivalent measures (i.e. they agree on which sets have measure zero/"are impossible"), we can rewrite this as $$ H_{0}=D(0,T)\mathbb{E}_{\mathbb{P}}\left[\frac{d\mathbb{Q}}{d\mathbb{P}}H_{T}\right] $$ where $\frac{d\mathbb{Q}}{d\mathbb{P}}$ is the Radon-Nikodym derivative of the measure $\mathbb{Q}$ w.r.t. $\mathbb{P}$, which is a function that "re-weights" the empirical measure so as to eliminate the "drift" (or risk) component of the payoff of the option and turns $H_{T}$ into a martingale such that $$ \mathbb{E}_{\mathbb{Q}}[H_{s}\lvert \{H_{\tau}\lvert\tau<s\}]=H_{s} $$ which basically means that the expected value of the payoff is constant under the equivalent measure $\mathbb{Q}$.
Now, this is a very simplified version of the theory but it should give you an idea that there are many different probability measures that exist and sometimes more than one that is useful.
In the real world, the parameters will usually be approximated emperically. Consider some weird non-fair die, then one can approximate the probabilities by rolling a bunch of times. In your example $\lambda$ is most likely also estimated based on data.
In the real world your mathematical model will rarely be exactly correct. The parameter values will not be exact - in fact, it will probably not even be a true exponential distribution. Instead, we get useful approximations to the real world with mathematical models.