exercise:F3b8de006a: 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>[-1,1]</math> with uniform distribution, and consider the quadratic equation <math display="block"> x^2 + Bx + C = 0\ . </math> Find the prob...") |
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Choose independently two numbers <math>B</math> and <math>C</math> ''at random'' from the interval <math>[-1,1]</math> with uniform distribution, and consider the quadratic equation | |||
interval <math>[-1,1]</math> with uniform distribution, and consider the quadratic | |||
equation | |||
<math display="block"> | <math display="block"> | ||
| Line 13: | Line 5: | ||
</math> | </math> | ||
Find the probability that the roots of this equation | Find the probability that the roots of this equation | ||
<ul><li> are both real. | <ul style="list-style-type:lower-alpha"><li> are both real. | ||
</li> | </li> | ||
<li> are both positive. | <li> are both positive. | ||
</li> | </li> | ||
</ul> | </ul> | ||
''Hints'': (a) requires <math>0 \leq B^2 - 4C</math>, | ''Hints'': (a) requires <math>0 \leq B^2 - 4C</math>, | ||
(b) requires <math>0 \leq B^2 - 4C</math>, <math>B \leq 0</math>, <math>0 \leq C</math>. | (b) requires <math>0 \leq B^2 - 4C</math>, <math>B \leq 0</math>, <math>0 \leq C</math>. | ||
Latest revision as of 23:25, 12 June 2024
Choose independently two numbers [math]B[/math] and [math]C[/math] at random from the interval [math][-1,1][/math] with uniform distribution, and consider the quadratic equation
[[math]]
x^2 + Bx + C = 0\ .
[[/math]]
Find the probability that the roots of this equation
- are both real.
- are both positive.
Hints: (a) requires [math]0 \leq B^2 - 4C[/math], (b) requires [math]0 \leq B^2 - 4C[/math], [math]B \leq 0[/math], [math]0 \leq C[/math].