This paper explores the application of Laplace's first law of errors to diffusive motion, particularly in the context of continuous time random walks (CTRW). The study investigates how the probability density function (PDF) of particle displacement, $ P(x,t) $, behaves under different jump length distributions. The central limit theorem (CLT) is shown to fail in capturing the behavior of diffusive processes in the tails, especially when the jump length distribution is super-exponential or sub-exponential. For super-exponential distributions, the large deviations theory is applied, which relates to the appearance of exponential tails in $ P(x,t) $. For sub-exponential distributions, the big jump principle is used to describe the spreading of particles. The study demonstrates that Laplace's first law can be applied for finite time, allowing rare events and the asymptotics of the large deviations rate function to be sampled for large length scales within a reasonably short measurement time. The paper analyzes the CTRW model with exponentially distributed waiting times and jump lengths following a specific PDF. It shows that for $ \beta > 1 $, the PDF $ P(x,t) $ exhibits exponential decay, consistent with Laplace's first law. For $ \beta < 1 $, the big jump principle applies, leading to sub-exponential behavior. The study also introduces the Edgeworth expansion to correct the CLT and provides a more accurate description of the PDF in different regimes. The paper discusses the transition between the big jump principle and large deviations theory, highlighting the importance of these frameworks in understanding diffusive processes. It also addresses the challenges of slow convergence in the asymptotic rate function (ARF) near the critical transition point $ \beta = 1 $, and the implications of non-exponential waiting times and varying dimensions on the results. The study concludes that Laplace's first law is not limited to the CTRW framework and has broader applications in various physical systems. The findings extend the utility of large deviations theory and the big jump principle beyond theoretical models, providing practical tools for analyzing particle dispersion in systems that deviate from normal distributions. The research underscores the importance of understanding rare events and non-Gaussian diffusion in biological, glassy, and active systems.This paper explores the application of Laplace's first law of errors to diffusive motion, particularly in the context of continuous time random walks (CTRW). The study investigates how the probability density function (PDF) of particle displacement, $ P(x,t) $, behaves under different jump length distributions. The central limit theorem (CLT) is shown to fail in capturing the behavior of diffusive processes in the tails, especially when the jump length distribution is super-exponential or sub-exponential. For super-exponential distributions, the large deviations theory is applied, which relates to the appearance of exponential tails in $ P(x,t) $. For sub-exponential distributions, the big jump principle is used to describe the spreading of particles. The study demonstrates that Laplace's first law can be applied for finite time, allowing rare events and the asymptotics of the large deviations rate function to be sampled for large length scales within a reasonably short measurement time. The paper analyzes the CTRW model with exponentially distributed waiting times and jump lengths following a specific PDF. It shows that for $ \beta > 1 $, the PDF $ P(x,t) $ exhibits exponential decay, consistent with Laplace's first law. For $ \beta < 1 $, the big jump principle applies, leading to sub-exponential behavior. The study also introduces the Edgeworth expansion to correct the CLT and provides a more accurate description of the PDF in different regimes. The paper discusses the transition between the big jump principle and large deviations theory, highlighting the importance of these frameworks in understanding diffusive processes. It also addresses the challenges of slow convergence in the asymptotic rate function (ARF) near the critical transition point $ \beta = 1 $, and the implications of non-exponential waiting times and varying dimensions on the results. The study concludes that Laplace's first law is not limited to the CTRW framework and has broader applications in various physical systems. The findings extend the utility of large deviations theory and the big jump principle beyond theoretical models, providing practical tools for analyzing particle dispersion in systems that deviate from normal distributions. The research underscores the importance of understanding rare events and non-Gaussian diffusion in biological, glassy, and active systems.