[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Suppose you are watching a radioactive source that emits particles at a rate described by the exponential density

[[math]] f(t) = \lambda e^{-\lambda t}\ , [[/math]]

where [math]\lambda = 1[/math], so that the probability [math]P(0,T)[/math] that a particle will appear in the next [math]T[/math] seconds is [math]P([0,T]) = \int_0^T\lambda e^{-\lambda t}\,dt[/math]. Find the probability that a particle (not necessarily the first) will appear

• within the next second.
• within the next 3 seconds.
• between 3 and 4 seconds from now.
• after 4 seconds from now.