{"id":3841,"date":"2014-10-06T14:37:35","date_gmt":"2014-10-06T19:37:35","guid":{"rendered":"http:\/\/www.livingreliability.com\/en\/?p=3841"},"modified":"2025-11-06T05:33:30","modified_gmt":"2025-11-06T10:33:30","slug":"conditional-probability-of-failure","status":"publish","type":"post","link":"https:\/\/www.livingreliability.com\/en\/posts\/conditional-probability-of-failure\/","title":{"rendered":"Conditional probability of failure"},"content":{"rendered":"

The most powerful\u00a0information sought by all maintenance engineers and managers boils down to\u00a0the conditional failure probability. It is the probability of an item failing in an\u00a0upcoming period of interest knowing that it is currently in an unfailed state. If you knew that the conditional probability of failure of a given\u00a0part or component were unusually high you could channel your\u00a0manpower to intervene propitiously, thereby preempting\u00a0the consequences of a failure in service while\u00a0avoiding waste of\u00a0resources and unnecessary downtime on items where\u00a0failure is not imminent.<\/em><\/p>\n

Other articles[1<\/a>]<\/sup> describe how to calculate\u00a0the conditional probability of failure of an item. In this article we’ll discuss\u00a0Conditional Probability first and then we’ll define the Conditional Probability of Failure.<\/p>\n

Conditional Probability<\/h3>\n

Let’s begin with a card experiment. A card is chosen at random from a standard\u00a0deck of 52 playing cards. Without replacing it, a second card is chosen. What is the probability that the first card chosen is a queen and the second card chosen is a jack? The events are said to be dependent because the probability of the second depends on the first.<\/p>\n

A. P(queen on first pick) = 4\/52
\nB. P(jack on 2nd pick given queen on 1st pick) = 4\/51, a higher\u00a0probability than 4\/52<\/p>\n

Then the probability that both events occur P(queen and jack)= (4\/52)(4\/51)=4\/663<\/p>\n

The probability of choosing a jack on the second pick given that a queen was chosen on the first pick is called a conditional probability<\/em>.\u00a0The conditional probability of an event B in relationship to an event A is the probability that event B occurs given that event A has already occurred. The notation for conditional probability is P(B|A) [2<\/a>]<\/sup>.<\/p>\n

When two events, A and B, are dependent, the probability of both occurring (denoted by “A\u2229B”<\/em>)\u00a0will, according to the card experiment, be the product of their probabilities, that is:<\/p>\n

P(A\u2229B) = P(A) \u00b7 P(B|A) ,\u00a0<\/em>or \"P(B|A)=\frac{P(A)\cap P(B)}{P(A)}\"<\/p>\n

When two events are dependent (the probability of one depends on the other’s occurrence) their probability areas intersect in a Venne graphical representation.<\/p>\n

\"venne\"<\/a><\/p>\n

Conditional Probability of Failure<\/h3>\n

Suppose\u00a0the two dependent events were:<\/p>\n

    \n
  1. X>t<\/em>, an\u00a0item survives to time t, X being the time of failure, and<\/li>\n
  2. t\u2264X\u2264t+\u0394t<\/em>, the\u00a0item fails in the interval between\u00a0t and t+\u0394t given event 1.<\/li>\n<\/ol>\n

    As in\u00a0the card experiment the probability of the second event depends on the first. Then the Conditional Probability of Failure is:<\/p>\n

    \"H(t)=P(t\leq X\leq t+\Delta t | X>t)=\frac{P(t\leq X\leq t+\Delta t)\cap P(X>t)}{P(X>t)}\"<\/p>\n

    It is the probability of failure in the interval\u00a0between\u00a0t and t+\u0394t (analogous to selecting a\u00a0jack\u00a0on the second\u00a0pick) given that the item has survived to time t (analogous to selecting a\u00a0queen on the first pick).<\/p>\n

    The article\u00a0here<\/a>\u00a0shows that the conditional failure probability is a special case of the conditional probability where the numerator\u00a0reduces\u00a0simply to P(t\u2264X\u2264t+\u0394t)<\/em>. So that:<\/p>\n

    \"H(t)=P(t\leq X\leq t+\Delta t | X>t)=\frac{P(t\leq X\leq t+\Delta t)}{P(X>t)}\"<\/p>\n\n

      \n\t
    1. [1]<\/sup><\/strong>See for example here<\/a> and here<\/a>.↩<\/a><\/li>\n\t
    2. [2]<\/sup><\/strong>pronounced as “The probability of event B given A”↩<\/a><\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"

      The most powerful\u00a0information sought by all maintenance engineers and managers boils down to\u00a0the conditional failure probability. It is the probability of an item failing in an\u00a0upcoming period of interest knowing that it is currently in an unfailed state. If you knew that the conditional probability of failure of a given\u00a0part or component were unusually high […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[89],"tags":[24],"class_list":["post-3841","post","type-post","status-publish","format-standard","hentry","category-theory-and-definitions","tag-conditional-failure-probability"],"_links":{"self":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/3841","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/comments?post=3841"}],"version-history":[{"count":1,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/3841\/revisions"}],"predecessor-version":[{"id":8722,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/3841\/revisions\/8722"}],"wp:attachment":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/media?parent=3841"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/categories?post=3841"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/tags?post=3841"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}