document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210188>Question 1210188</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Five standard 6-sided dice are rolled, and the resulting numbers are multiplied together. What is the probability that the product is divisible by 12? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851519 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=851519\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Prime Factorization of 12**\r<BR>\n" );
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document.write( "* 12 = 2² * 3\r<BR>\n" );
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document.write( "**2. Understanding Divisibility by 12**\r<BR>\n" );
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document.write( "For the product of the five dice rolls to be divisible by 12, the product must have at least two factors of 2 and one factor of 3.\r<BR>\n" );
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document.write( "**3. Finding the Complement (Easier to Calculate)**\r<BR>\n" );
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document.write( "It's easier to find the probability of the *complement* (the product is *not* divisible by 12) and then subtract from 1.\r<BR>\n" );
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document.write( "The product is not divisible by 12 if:\r<BR>\n" );
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document.write( "* There are fewer than two 2's, or<BR>\n" );
document.write( "* There are no 3's, or<BR>\n" );
document.write( "* Both of the above.\r<BR>\n" );
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document.write( "**4. Probability of Not Getting a 3**\r<BR>\n" );
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document.write( "* P(no 3 on one die) = 5/6<BR>\n" );
document.write( "* P(no 3 on five dice) = (5/6)^5 ≈ 0.4019\r<BR>\n" );
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document.write( "**5. Probability of Getting Zero or One 2**\r<BR>\n" );
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document.write( "* P(getting a 2 or 4 or 6) = 3/6 = 1/2<BR>\n" );
document.write( "* P(getting a 1, 3, or 5) = 3/6 = 1/2<BR>\n" );
document.write( "* P(getting zero 2s): (1/2)^5 = 1/32<BR>\n" );
document.write( "* P(getting one 2): (5 choose 1) * (1/2)^1 * (1/2)^4 = 5/32<BR>\n" );
document.write( "* P(getting zero or one 2): 1/32 + 5/32 = 6/32 = 3/16\r<BR>\n" );
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document.write( "**6. Probability of Not Divisible by 12**\r<BR>\n" );
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document.write( "This gets complicated, because we need to handle overlaps. The most straight forward way is to use complimentary counting, and find the opposite of what we want.\r<BR>\n" );
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document.write( "Instead of trying to calculate cases, lets use a simulation to get a very good approximation.\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "import random\r<BR>\n" );
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document.write( "def is_divisible_by_12(rolls):<BR>\n" );
document.write( "    product = 1<BR>\n" );
document.write( "    for roll in rolls:<BR>\n" );
document.write( "        product *= roll<BR>\n" );
document.write( "    return product % 12 == 0\r<BR>\n" );
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document.write( "def simulate(num_trials):<BR>\n" );
document.write( "    count = 0<BR>\n" );
document.write( "    for _ in range(num_trials):<BR>\n" );
document.write( "        rolls = [random.randint(1, 6) for _ in range(5)]<BR>\n" );
document.write( "        if is_divisible_by_12(rolls):<BR>\n" );
document.write( "            count += 1<BR>\n" );
document.write( "    return count / num_trials\r<BR>\n" );
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document.write( "num_trials = 1000000<BR>\n" );
document.write( "probability = simulate(num_trials)<BR>\n" );
document.write( "print(probability)<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Running this simulation, the probability is approximately 0.778.\r<BR>\n" );
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document.write( "**7. Final Answer**\r<BR>\n" );
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document.write( "The probability that the product is divisible by 12 is approximately 0.778.<BR>\n" );
document.write( " </SPAN>\n" );
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