document.write( "Question 1105966: I do not know how they got the answer\r
\n" ); document.write( "\n" ); document.write( "Three dice are rolled. Find each probability:\r
\n" ); document.write( "\n" ); document.write( "a) One of the rolls is a 6, given all rolls are even. ans. 12/27\r
\n" ); document.write( "\n" ); document.write( "b) One of the rolls is a 3, given two rolls are even. ans. 4/9\r
\n" ); document.write( "\n" ); document.write( "c) At least two of the rolls are even, given all three rolls are the same. ans. 1/2 \n" ); document.write( "
Algebra.Com's Answer #720911 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
Here is my conclusion:
\n" ); document.write( "a) If all rolls are even, there is exactly one roll that is a 6 in \"highlight%2812%2F27=4%2F9%29\" of the equally possible cases.
\n" ); document.write( "b) If exactly two rolls are even, there is exactly one roll that is a 3 in \"highlight%283%2F9=1%2F3%29\" of all equally possible cases.
\n" ); document.write( "The only place where I get a \"4%2F9\" is the probability of rolling exactly one 3, when exactly two of the numbers rolled were odd.
\n" ); document.write( "c) At least two of the rolls are even, If all three rolls are the same, all rolls are even in 1/2 of all 6 possible cases, and all rolls are odd in the other 1/2 of all 6 possible cases. That means that \"highlight%283%2F6=1%2F2%29\" of the 6 cases where all rolls are the same have at least two of the rolls that are even (and all rolls are even in those cases.
\n" ); document.write( "
\n" ); document.write( "How I got those answers:
\n" ); document.write( "a) If a roll of one die is an even number, a 6 is as likely as a 4 or a 2,
\n" ); document.write( "so there are \"red%283%29\" equally likely possibilities for each die.
\n" ); document.write( "
\n" ); document.write( "One way to \"show your work\" is to say that with an even roll,
\n" ); document.write( "the probability of 6 is is \"P%286%29=1%2F3\" and \"P%28not6%29=2%2F3\" .
\n" ); document.write( "For one die, those probabilities are represented by the corfficients of \"s\" and \"n\" in \"%281%2F3%29s%2B%282%2F3%29n\" .
\n" ); document.write( "You would then say that applying binomial distribution probability,
\n" ); document.write( "with \"green%283%29\" dice being rolled, the probability of exactly \"1\" 6 is
\n" ); document.write( "the coefficient in the term with \"s%5E1=s\" in the expansion of
\n" ); document.write( "\"%28%281%2F3%29s%2B%282%2F3%29n%29%5Egreen%283%29\" .
\n" ); document.write( "That expansion is
\n" ); document.write( "\"%28%281%2F3%29s%2B%282%2F3%29n%29%5Egreen%283%29\"\"%22=%22\"\"%28%281%2F3%29s%29%5E3\"\"%22%2B%22\"\"3%2A%28%281%2F3%29s%29%5E2%28%282%2F3%29n%29%29\"\"%22%2B%22\"\"3%28%281%2F3%29s%29%28%282%2F3%29n%29%5E2\"\"%22%2B%22\"\"%28%282%2F3%29n%29%5E3\"\"%22=%22\"\"%281%2F9%29s%5E3\"\"%22%2B%22\"\"%282%2F9%29s%5E2%2An%29%29\"\"%22%2B%22\"\"%284%2F9%29sn%5E2\"\"%22%2B%22\"\"%288%2F27%29s%5E3\" .
\n" ); document.write( "
\n" ); document.write( "Another way would involve calculating counts (or literall ycounting):
\n" ); document.write( "With three dice, you would have
\n" ); document.write( "\"red%283%29%2Ared%283%29%2Ared%283%29=red%283%29%5E3=27\" equally likely possibilities (outcomes).
\n" ); document.write( "You could similarly calculate the different kinds of outcomes as
\n" ); document.write( "\"1\" of those \"27\" equally likely outcomess would be a 6 from all 3 dice (three 6's),
\n" ); document.write( "\"6=3%2A2\" would be one of the \"3\" even rolls being one of the \"2\" non-6 even numbers (two 6's),
\n" ); document.write( "\"8=2%5E3\" would be all \"3\" rolls being one of the \"2\" non-6 even numbers (no 6\"s),
\n" ); document.write( "and the remaining \"27-1-6-8=12\" would be exactly one 6 rolled.
\n" ); document.write( "You can also calculate that \"12\" (the number of \"favorable outcomes\") as
\n" ); document.write( "\"3\" places to put the one 6,
\n" ); document.write( "times \"2%2A2=4\" possibilities for the two other non-6 even numbers rolled.
\n" ); document.write( "
\n" ); document.write( "Dealing with such a small total number of equally likely outcomes,
\n" ); document.write( "you could literally count them.
\n" ); document.write( "You can list them as 3 digit numbers,
\n" ); document.write( "where each digit position would represent the result of rolling one specific die.
\n" ); document.write( "Then you could literally count outcomes.
\n" ); document.write( "You could distinguish the three dice by colors
\n" ); document.write( "(as the red die, the white die, and the blue dice, listed in that order),
\n" ); document.write( "or you could distinguish the dice by when and where they were rolled.
\n" ); document.write( "The sequences of 3 (all even) numbers rolled
\n" ); document.write( "(by the first second and third die rolled, in that order) can be listed as
\n" ); document.write( "
\n" ); document.write( "There are \"27\" outcomes,
\n" ); document.write( "and \"12\" of them have exactly one 6,
\n" ); document.write( "so the probability of getting exactly one 6 when the 3 dice rolls are even is
\n" ); document.write( "\"12%2F27=4%2F9\" .
\n" ); document.write( "
\n" ); document.write( "b) If two of the 3 rolls are even, and the other roll is odd,
\n" ); document.write( "the situation is the same for any die the odd roll comes from,
\n" ); document.write( "and for any arrangement of even numbers rolled by the other two dice.
\n" ); document.write( "There are 3 possibilities for the odd roll: 1, 3, or 5.
\n" ); document.write( "Each odd number will be in \"1%2F3\" of the two-even-one-odd rolls.
\n" ); document.write( "We do not even need to calculate or count outcomes, but here it goes.
\n" ); document.write( "There are \"3\" odd numbers that could be the odd roll.
\n" ); document.write( "There are \"red%283%29\" dice that could be the odd roll.
\n" ); document.write( "There are \"green%283%29\" possible even numbers that could be rolled
\n" ); document.write( "for each of the other \"2\" dice.
\n" ); document.write( "That makes for \"3%2Ared%283%29%2Agreen%283%29%5E2=81\" different outcomes.
\n" ); document.write( "Of those, \"3%2Ared%283%29%2Agreen%283%29%5E2=27\" would include rolling a 3.
\n" ); document.write( "\"27%2F81=1%2F3\" .
\n" ); document.write( "
\n" ); document.write( "What about counting all outcomes to make sure that is right?
\n" ); document.write( "I had my computer do it with a spreadsheet .
\n" ); document.write( "Of the \"6%5E3=216\" possible outcomes of rolling 3 dice,
\n" ); document.write( "There were \"27\" different, equally likely, all-odd outcomes,
\n" ); document.write( "\"27\" different, equally likely, all-even outcomes,
\n" ); document.write( "\"81\" different, equally likely, two-even-one-odd outcomes,
\n" ); document.write( "and \"81\" different, equally likely, two-odd-one-even outcomes.
\n" ); document.write( "Of the \"81\" two-even-one-odd outcomes, \"27\" included rolling one 3.
\n" ); document.write( "
\n" ); document.write( "c) All three rolls being the same includes 6 different, equally likely outcomes:
\n" ); document.write( "all rolls are 1, all are 2, all are 3, all are 4, all are 5, and all are 6.
\n" ); document.write( "In \"3\" of those \"6\" cases, the number on all 3 dice is even,
\n" ); document.write( "and in for the other 3 all-rolls-the-same, the number on all 3 dice is
\n" ); document.write( "That is, in \"1%2F2\" of the outcomes considered, 3 rolls are even,
\n" ); document.write( "and in \"1%2F2\" of the outcomes considered, 3 rolls are odd.
\n" ); document.write( "In other words, in \"1%2F2\" of the outcomes considered, none of the numbers rolled is even,
\n" ); document.write( "and the other \"1%2F2\" of the outcomes considered,
\n" ); document.write( "at least 2 of the numbers rolled are even, because all 3 are. \n" ); document.write( "
\n" );