document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166442>Question 1166442</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?\r<BR>\n" );
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document.write( "Please help, thanks  \r<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 #852756 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=852756\">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 **99.44%** (or **0.9944**).\r<BR>\n" );
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document.write( "This is a **Bayes' Theorem** problem, where we update our initial belief (prior probability) based on new evidence (the green light). \r<BR>\n" );
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document.write( "## 🔬 Probability Setup\r<BR>\n" );
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document.write( "Let $O$ be the event that the door **Should Open** (i.e., it is unlocked).<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( "* $P(O)$ (Probability the door is unlocked) $= 90\% = 0.90$<BR>\n" );
document.write( "* $P(L)$ (Probability the door is locked) $= 1 - P(O) = 1 - 0.90 = 0.10$\r<BR>\n" );
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document.write( "### 2. Likelihoods (System Accuracy)\r<BR>\n" );
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document.write( "* **True Positive Rate ($P(G|O)$):** Probability of a green light when the door *should* open.<BR>\n" );
document.write( "    $$P(G|O) = 98\% = 0.98$$<BR>\n" );
document.write( "* **False Positive Rate ($P(G|L)$):** Probability of a green light when the door *should not* open (system error).<BR>\n" );
document.write( "    $$P(G|L) = 5\% = 0.05$$\r<BR>\n" );
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document.write( "### 3. Goal\r<BR>\n" );
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document.write( "We want to find $P(O|G)$, the probability that the door is actually unlocked (and will open) given that we see a green light.\r<BR>\n" );
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document.write( "## 🧮 Applying Bayes' Theorem\r<BR>\n" );
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document.write( "$$P(O|G) = \frac{P(G|O) \cdot P(O)}{P(G)}$$\r<BR>\n" );
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document.write( "### Step 1: Calculate the Total Probability of a Green Light ($P(G)$)\r<BR>\n" );
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document.write( "The green light can occur in two ways: correctly (True Positive) or incorrectly (False Positive).\r<BR>\n" );
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document.write( "$$P(G) = P(\text{Green and Open}) + P(\text{Green and Locked})$$<BR>\n" );
document.write( "$$P(G) = [P(G|O) \cdot P(O)] + [P(G|L) \cdot P(L)]$$<BR>\n" );
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 the system showing a green light is $88.7\%$.\r<BR>\n" );
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document.write( "### Step 2: Calculate the Posterior Probability $P(O|G)$\r<BR>\n" );
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document.write( "$$P(O|G) = \frac{P(\text{Green and Open})}{P(\text{Total Green})}$$<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( "Rounding to four decimal places, the probability the door will open is **$0.9944$**. </SPAN>\n" );
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