# Fault Tree Top Event Probability Calculator: OR and AND Gates with Redundancy > Source: https://netray.co/tools/fault-tree-top-event-probability-calculator > Publisher: Netray (netray.co), AI and integration engineering for ERP and manufacturing systems. > Section: Free Tools ## Short answer Computes the probability of a top event from three branches: a single event, a redundant pair joined by an AND gate, and another single event, all joined by an OR gate. ## Category quality ## Audience - Reliability engineer - Safety engineer - Quality engineer ## Tool **Kind:** calculator **Inputs:** #### Event A probability **Key:** p_a **Type:** number **Default:** 2 **Min:** 0 **Max:** 100 **Step:** 0.1 **Unit:** % **Help:** Single point failure feeding the OR gate. #### Event B probability **Key:** p_b **Type:** number **Default:** 5 **Min:** 0 **Max:** 100 **Step:** 0.1 **Unit:** % **Help:** First leg of the redundant pair. #### Event C probability **Key:** p_c **Type:** number **Default:** 5 **Min:** 0 **Max:** 100 **Step:** 0.1 **Unit:** % **Help:** Second leg of the redundant pair. B and C must both fail. #### Event D probability **Key:** p_d **Type:** number **Default:** 1 **Min:** 0 **Max:** 100 **Step:** 0.1 **Unit:** % **Help:** Another single point failure feeding the OR gate. **Outputs:** #### A as a fraction **Key:** a **Expr:** p_a / 100 **Format:** decimal #### B as a fraction **Key:** b **Expr:** p_b / 100 **Format:** decimal #### C as a fraction **Key:** c **Expr:** p_c / 100 **Format:** decimal #### D as a fraction **Key:** d **Expr:** p_d / 100 **Format:** decimal #### Redundant pair fails (B AND C) **Key:** bc **Expr:** b * c * 100 **Format:** percent **Description:** Product of the two probabilities, valid if B and C are independent. #### Top event probability **Key:** top **Expr:** (1 - (1 - a) * (1 - b * c) * (1 - d)) * 100 **Format:** percent **Primary:** true **Description:** Exact probability for independent events: 1 minus the product of the complements. #### Top event without redundancy (B OR C) **Key:** no_red **Expr:** (1 - (1 - a) * (1 - b) * (1 - c) * (1 - d)) * 100 **Format:** percent **Description:** If B and C were both single points of failure. #### Reduction from redundancy **Key:** reduction **Expr:** no_red - top **Format:** percent **Primary:** true **Description:** Percentage points of top event probability removed by the redundant pair. **Chart:** #### Top event probability versus event B probability **Type:** line **X:** p_b **From:** 0 **To:** 30 **Steps:** 30 **Series:** - top - no_red **Disclaimer:** Assumes statistically independent basic events and probabilities over the same mission time or demand. Common-cause failures can make a redundant pair far less effective than shown. Not a substitute for a formal safety analysis. ## Content **Intro:** Fault tree analysis starts from an unwanted top event and works down to the basic events that cause it. This calculator handles a common small structure: two single points of failure plus a redundant pair, all feeding one top event. It shows how much the redundancy is worth, and how fast that benefit shrinks as the pair gets less reliable. **Sections:** #### How to use it **Body:** Enter the probability of each basic event over the same period, such as one mission or one demand, in percent. Events A and D are single points of failure: if either happens, the top event happens. Events B and C are a redundant pair: the top event occurs through that branch only if both fail. The calculator joins the three branches with an OR gate and shows the top event probability, the same figure if the pair were not redundant, and the reduction. #### The math behind it **Body:** For independent events, an AND gate multiplies probabilities: P(B and C) = P(B) x P(C). An OR gate is one minus the product of the complements: P(A or X or D) = 1 - (1 - P(A))(1 - P(X))(1 - P(D)). For small probabilities this is close to the sum, which is the rare-event approximation used in many reports. Standards such as IEC 61025 describe the method. In larger trees, minimal cut sets are found first, and the top event probability is the union of those sets. **Bullets:** - AND gate: multiply the input probabilities - OR gate: one minus the product of (1 - p) - Rare-event approximation: add the cut set probabilities - Check that all probabilities share the same time basis #### Worked example and common mistakes **Body:** With A at 2%, B and C at 5% each and D at 1%, the redundant pair fails together only 0.25% of the time. The top event is about 3.2%, against about 12.5% if B and C were each single points of failure. The benefit looks large, but it depends on independence. If B and C share a power supply or a software defect, the true pair probability may be orders of magnitude higher. Mixing per-hour and per-demand figures is another common error. **Cta:** If you want failure rates and downtime events pulled from your maintenance and quality systems to feed trees like this, Netray can discuss a read-only approach with your data. **Faq:** #### Why multiply probabilities for an AND gate? **Answer:** If both events must occur and they are independent, the chance of both is the product of the chances. Five percent and five percent give 0.25 percent. This holds only when one failure does not make the other more likely. #### What is common cause failure? **Answer:** A single root cause that defeats redundant items at once, such as shared power, a shared environment, a design flaw or the same maintenance error. It makes AND gate math too optimistic and must be modelled separately. #### Can I use failure rates instead of probabilities? **Answer:** Convert them first. For a constant failure rate lambda and a mission time t, the probability of failure is about 1 minus exp(-lambda x t). Then enter that figure as a percent so all events share the same time basis. #### Is this enough for a safety case? **Answer:** No. Real fault trees can have many gates, repeated events and dependencies, and should be built with dedicated software and reviewed against IEC 61025 and your sector standard. This tool is for quick checks and teaching. ## Search terms fault tree analysis calculator, fta probability calculator, and gate or gate probability, top event probability, redundancy reliability calculator, fault tree example, fta calculation example, system failure probability calculator --- Talk to an engineer: https://netray.co/contact?intent=erp_ai | Machine catalog: https://netray.co/agent/catalog.json