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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">With high probability</span></span>
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, an event that occurs <b>with high probability</b> (often shortened to <b>w.h.p.</b> or <b>WHP</b>) is one whose <a href="Probability" title="Probability">probability</a> depends on a certain number <i>n</i> and goes to 1 as <i>n</i> goes to infinity, i.e. the probability of the <a href="Event_(probability_theory)" title="Event (probability theory)">event</a> occurring can be made as close to 1 as desired by making <i>n</i> big enough.
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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>
<p>The term WHP is especially common in <a href="Computer_science" title="Computer science">computer science</a>, in the analysis of <a href="Probabilistic_algorithm" class="mw-redirect" title="Probabilistic algorithm">probabilistic algorithms</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> For example, consider a certain probabilistic algorithm on a graph with <i>n</i> nodes. If the probability that the algorithm returns the correct answer is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-1/n}">
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<annotation encoding="application/x-tex">{\displaystyle 1-1/n}</annotation>
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</math></span><img src="./88e1d11b673218dccfccbf15915c62986e0f6235.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.722ex; height:2.843ex;" alt="{\displaystyle 1-1/n}" loading="lazy"></span>, then when the number of nodes is very large, the algorithm is correct with a probability that is very near 1. This fact is expressed shortly by saying that the algorithm is correct WHP.
</p><p>Some examples where this term is used are:
</p>
<ul><li><a href="Miller%E2%80%93Rabin_primality_test" title="Miller–Rabin primality test">Miller–Rabin primality test</a>: a probabilistic algorithm for testing whether a given number <i>n</i> is prime or composite. If <i>n</i> is composite, the test will detect <i>n</i> as composite WHP. There is a small chance that we are unlucky and the test will think that <i>n</i> is prime. But, the probability of error can be reduced indefinitely by running the test many times with different randomizations.</li>
<li><a href="Freivalds'_algorithm" title="Freivalds' algorithm">Freivalds' algorithm</a>: a randomized algorithm for verifying matrix multiplication. It runs faster than deterministic algorithms WHP.</li>
<li><a href="Treap" title="Treap">Treap</a>: a randomized binary search tree. Its height is logarithmic WHP. <a href="Fusion_tree" title="Fusion tree">Fusion tree</a> is a related data structure.</li>
<li><a href="Online_codes" title="Online codes">Online codes</a>: randomized codes which allow the user to recover the original message WHP.</li>
<li><a href="BQP" title="BQP">BQP</a>: a complexity class of problems for which there are polynomial-time quantum algorithms which are correct WHP.</li>
<li><a href="Probably_approximately_correct_learning" title="Probably approximately correct learning">Probably approximately correct learning</a>: A process for machine-learning in which the learned function has low generalization-error WHP.</li>
<li><a href="Gossip_protocol" title="Gossip protocol">Gossip protocols</a>: a <a href="Communication_protocol" title="Communication protocol">communication protocol</a> used in <a href="Distributed_computing" title="Distributed computing">distributed systems</a> to reliably deliver messages to the whole cluster using a constant amount of network resources on each node and ensuring no single point of failure.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Randomized_algorithm" title="Randomized algorithm">Randomized algorithm</a></li>
<li><a href="Almost_surely" title="Almost surely">Almost surely</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFMitzenmacherUpfal2012" class="citation book cs1">Mitzenmacher, Michael; Upfal, Eli (2012). <i>Probability and computing: randomized algorithms and probabilistic analysis</i>. Cambridge: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-83540-4</bdi>.</cite></span>
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<ul><li><cite id="CITEREFMétivierRobsonSaheb-DjahromiZemmari2010" class="citation journal cs1">Métivier, Y.; Robson, J. M.; Saheb-Djahromi, N.; Zemmari, A. (2010). "An optimal bit complexity randomized distributed MIS algorithm". <i>Distributed Computing</i>. <b>23</b> (<span class="nowrap">5–</span>6): 331. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00446-010-0121-5">10.1007/s00446-010-0121-5</a>.</cite></li>
<li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://dcg.ethz.ch/lectures/podc_allstars/lecture/chapter7.pdf">"Principles of Distributed Computing (lecture 7)"</a> <span class="cs1-format">(PDF)</span>. ETH Zurich<span class="reference-accessdate">. Retrieved <span class="nowrap">21 February</span> 2015</span>.</cite></li></ul>
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