document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166560>Question 1166560</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Due to the high incidence of crime, a company is giving its employees new and secured access to the company.  Instead of leaving the doors open, they are installing a card system.  To open the door, employees must insert a card into a slot.  If a green light comes on, it is okay to turn the handle and open the door; if a yellow light comes on it indicates the door is locked from inside and you cannot enter.  Suppose that 90% of the time when the card is inserted, the door should open because it is not locked from the inside.  When the door should open, the system makes errors 2% of the time.  That is, the green light will appear 98% of the time.  When the door should not open, the system makes errors 5% of the time (green light appears).  Suppose you inserted the card and the light is green, what is the probability the door will open?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852744 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2030)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852744\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The probability that the door will open, given that the light is green, is approximately **99.71%**.\r<BR>\n" );
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document.write( "This is a classic problem that can be solved using **Bayes' Theorem**.\r<BR>\n" );
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document.write( "## 🧠 Setup and Given Probabilities\r<BR>\n" );
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document.write( "Let $O$ be the event that the door **Should Open** (i.e., it is not locked).<BR>\n" );
document.write( "Let $L$ be the event that the door **Should Not Open** (i.e., it is locked).<BR>\n" );
document.write( "Let $G$ be the event that the **Green Light** appears.\r<BR>\n" );
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document.write( "### 1. Prior Probabilities (What *should* happen)\r<BR>\n" );
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document.write( "* $P(O) = 0.90$ (90% of the time the door should open).<BR>\n" );
document.write( "* $P(L) = 1 - P(O) = 1 - 0.90 = 0.10$ (10% of the time the door should not open).\r<BR>\n" );
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document.write( "### 2. Conditional Probabilities (Accuracy and Error Rates)\r<BR>\n" );
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document.write( "* **Case 1: Door Should Open ($O$)**<BR>\n" );
document.write( "    * System is correct (Green light $G$ appears): $P(G|O) = 0.98$.<BR>\n" );
document.write( "    * System makes an error (Yellow light appears): $1 - 0.98 = 0.02$.\r<BR>\n" );
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document.write( "* **Case 2: Door Should Not Open ($L$)**<BR>\n" );
document.write( "    * System makes an error (Green light $G$ appears): $P(G|L) = 0.05$.<BR>\n" );
document.write( "    * System is correct (Yellow light appears): $1 - 0.05 = 0.95$.\r<BR>\n" );
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document.write( "### 3. Goal\r<BR>\n" );
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document.write( "We want to find the probability that the door **will open** (event $O$) given that the **light is green** (event $G$). This is the posterior probability, $P(O|G)$.\r<BR>\n" );
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document.write( "## 🧮 Applying Bayes' Theorem\r<BR>\n" );
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document.write( "Bayes' Theorem states:<BR>\n" );
document.write( "$$P(O|G) = \frac{P(G|O) \cdot P(O)}{P(G)}$$\r<BR>\n" );
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document.write( "We first need to find the **Total Probability of the Green Light ($P(G)$)** using the Law of Total Probability:<BR>\n" );
document.write( "$$P(G) = P(G|O) \cdot P(O) + P(G|L) \cdot P(L)$$\r<BR>\n" );
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document.write( "### Step 1: Calculate $P(G)$\r<BR>\n" );
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document.write( "$$P(G) = (0.98) \cdot (0.90) + (0.05) \cdot (0.10)$$<BR>\n" );
document.write( "$$P(G) = 0.882 + 0.005$$<BR>\n" );
document.write( "$$P(G) = 0.887$$\r<BR>\n" );
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document.write( "The overall probability of seeing a green light is $88.7\%$.\r<BR>\n" );
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document.write( "### Step 2: Calculate $P(O|G)$\r<BR>\n" );
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document.write( "Now substitute $P(G)$ into Bayes' Theorem:<BR>\n" );
document.write( "$$P(O|G) = \frac{0.98 \cdot 0.90}{0.887}$$<BR>\n" );
document.write( "$$P(O|G) = \frac{0.882}{0.887}$$<BR>\n" );
document.write( "$$P(O|G) \approx 0.99436$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## ⚠️ Interpretation of the Question\r<BR>\n" );
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document.write( "The question asks for the probability the door **will open** given the light is green. The event \"the door should open\" ($O$) is equivalent to the event \"the door will open\" if the system is reliable.\r<BR>\n" );
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document.write( "**If the question assumes the system is $100\%$ reliable when the light is green (which it usually does in such contexts, where $G$ means permission to open):**<BR>\n" );
document.write( "The probability the door **should open** ($O$) given the green light ($G$) is $P(O|G) \approx \mathbf{0.9944}$.\r<BR>\n" );
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document.write( "**If the question assumes the door can *only* open if the *system allows it* (i.e., the system is a perfect intermediary):**<BR>\n" );
document.write( "$P(\text{Door will open }| G) = 1$<BR>\n" );
document.write( "This is not the intended statistical question.\r<BR>\n" );
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document.write( "The intended answer is the probability that the door is unlocked when the light is green:\r<BR>\n" );
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document.write( "$$P(\text{Door will open } | G) \approx \mathbf{0.9944}$$ </SPAN>\n" );
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