exercise:246cabd63e: Difference between revisions
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(Created page with "<div class="d-none"><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></div> Choose independently two numbers <math>B</math> and <math>C</math> ''at random'' from the interval <math>[0,1]</math> with uniform density. Note that the point <math>(B,C)</math> is then chosen ''at random'' in the unit square. Find the probabil...") |
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\newcommand{\mathds}{\mathbb}</math></div> Choose independently two numbers <math>B</math> and <math>C</math> | \newcommand{\mathds}{\mathbb}</math></div> Choose independently two numbers <math>B</math> and <math>C</math> ''at random'' from the interval <math>[0,1]</math> with uniform density. Note that the point <math>(B,C)</math> is then chosen ''at random'' in the unit square. Find the probability that | ||
''at random'' from the interval <math>[0,1]</math> with uniform density. Note that | |||
the point <math>(B,C)</math> is then chosen ''at random'' in the unit square. Find the | <ul style="list-style-type:lower-alpha"><li> <math>B + C < 1/2</math>. | ||
probability that | |||
<ul><li> <math>B + C < 1/2</math>. | |||
</li> | </li> | ||
<li> <math>BC < 1/2</math>. | <li> <math>BC < 1/2</math>. | ||
Latest revision as of 21:38, 12 June 2024
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Choose independently two numbers [math]B[/math] and [math]C[/math] at random from the interval [math][0,1][/math] with uniform density. Note that the point [math](B,C)[/math] is then chosen at random in the unit square. Find the probability that
- [math]B + C \lt 1/2[/math].
- [math]BC \lt 1/2[/math].
- [math]|B - C| \lt 1/2[/math].
- [math]\max\{B,C\} \lt 1/2[/math].
- [math]\min\{B,C\} \lt 1/2[/math].
- [math]B \lt 1/2[/math] and [math]1 - C \lt 1/2[/math].
- conditions (c) and (f) both hold.
- [math]B^2 + C^2 \leq 1/2[/math].
- [math](B - 1/2)^2 + (C - 1/2)^2 \lt 1/4[/math].