exercise:173c44e9bd: 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> Let <math>j</math> and <math>n</math> be positive integers, with <math>j \le n</math>. An experiment consists of choosing, at random, a <math>j</math>-tuple of ''positive'' integers whose sum is at most <math>n</math>. <ul><li> Find the size of t...") |
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Let <math>j</math> and <math>n</math> be positive integers, with <math>j \le n</math>. An experiment consists of choosing, at random, a <math>j</math>-tuple of ''positive'' integers | |||
experiment consists of choosing, at random, a <math>j</math>-tuple of ''positive'' integers | |||
whose sum is at most <math>n</math>. | whose sum is at most <math>n</math>. | ||
<ul><li> Find the size of the sample space. '' Hint'': Consider <math>n</math> indistinguishable | <ul style="list-style-type:lower-alpha"><li> Find the size of the sample space. '' Hint'': Consider <math>n</math> indistinguishable | ||
balls placed in a row. Place <math>j</math> markers between consecutive pairs of balls, with no | balls placed in a row. Place <math>j</math> markers between consecutive pairs of balls, with no | ||
two markers between the same pair of balls. (We also allow one of the <math>n</math> markers to be | two markers between the same pair of balls. (We also allow one of the <math>n</math> markers to be | ||
Latest revision as of 00:13, 13 June 2024
Let [math]j[/math] and [math]n[/math] be positive integers, with [math]j \le n[/math]. An experiment consists of choosing, at random, a [math]j[/math]-tuple of positive integers whose sum is at most [math]n[/math].
- Find the size of the sample space. Hint: Consider [math]n[/math] indistinguishable balls placed in a row. Place [math]j[/math] markers between consecutive pairs of balls, with no two markers between the same pair of balls. (We also allow one of the [math]n[/math] markers to be placed at the end of the row of balls.) Show that there is a 1-1 correspondence between the set of possible positions for the markers and the set of [math]j[/math]-tuples whose size we are trying to count.
- Find the probability that the [math]j[/math]-tuple selected contains at least one 1.