document.write( "Question 1156743: A box contains three coins, two of them fair and one two-headed. A coin is
\n" ); document.write( "selected at random and tossed. If heads appear the coin is tossed again; if
\n" ); document.write( "tails appear then another coin is selected from the two remaining coins and
\n" ); document.write( "tossed.
\n" ); document.write( "(i) find the probability that heads appear twice.
\n" ); document.write( "(ii) if the same coin is tossed twice, find the probability that it is two -headed coin.
\n" ); document.write( "(iii) find the probability that tails appear twice. \n" ); document.write( "
Algebra.Com's Answer #779677 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
uestion 1156743\r
\n" ); document.write( "\n" ); document.write( "A box contains three coins, two of them fair and one two-headed. A coin is
\n" ); document.write( "selected at random and tossed. If heads appear the coin is tossed again; if
\n" ); document.write( "tails appear then another coin is selected from the two remaining coins and
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\n" ); document.write( "\n" ); document.write( "(i) find the probability that heads appear twice.
\n" ); document.write( "(ii) if the same coin is tossed twice, find the probability that it is the
\n" ); document.write( "two-headed coin.
\n" ); document.write( "(iii) find the probability that tails appear twice.
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document.write( "In each case we select a coin, toss a coin, select a coin, then toss a coin.\r\n" );
document.write( "Sometimes it is certain (probability = 1) which coin we select, sometimes not.\r\n" );
document.write( "Sometimes it's certain that we'll get a head, (probability = 1), sometimes not.\r\n" );
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document.write( "There are six cases.  I can't make a tree diagram here, but you can and should.\r\n" );
document.write( "Here are the six cases:\r\n" );
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document.write( "    Select(Prob.)|Toss(Prob.)  |Select(Prob.)| Toss(Prob.)| Probability of possibility \r\n" );
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document.write( "1.   Fair(2/3)   |   Head(1/2) |   Fair(1)   |  Head(1/2) |(2/3)(1/2)(1)(1/2)   = 1/6\r\n" );
document.write( "2.       \"       |       \"     |      \"      |  Tail(1/2) |(2/3)(1/2)(1)(1/2)   = 1/6 \r\n" );
document.write( "3.       \"       |   Tail(1/2) |   Fair(1/2) |  Head(1/2) |(2/3)(1/2)(1/2)(1/2) = 1/12\r\n" );
document.write( "4.       \"       |       \"     |      \"      |  Tail(1/2) |(2/3)(1/2)(1/2)(1/2) = 1/12 \r\n" );
document.write( "5.       \"       |       \"     | 2headed(1/2)|  Head(1)   |(2/3)(1/2)(1/2)(1)   = 1/6\r\n" );
document.write( "6. 2headed(1/3)  |    Head(1)  |  2headed(1) |  Head(1)   |(1/3)(1)(1)(1)       = 1/3\r\n" );
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document.write( "Notice that the sum of the probabilities of all six cases is 1, i.e., \r\n" );
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document.write( "1/6+1/6+1/12+1/12+1/6+1/3 = 2/12+2/12+1/12+1/12+2/12+4/12 = 12/12 = 1. \r\n" );
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(i) find the probability that heads appear twice.
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document.write( "P(case 1 OR case 6) = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2\r\n" );
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(ii) if the same coin is tossed twice, find the probability that it
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document.write( "P(2-headed|tossed twice) = P(case 6|case 1 or case 6) =\r\n" );
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document.write( " P(case 6 AND case 1 or 6)       P(case 6)        1/3\r\n" );
document.write( "--------------------------- = ---------------- = ----- = (1/3)∙(2/1) = 2/3\r\n" );
document.write( "    P(case 1 OR case 6)        1/2 [from (i)]     1/2 \r\n" );
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(iii) find the probability that tails appear twice.
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document.write( "P(case 4) = 1/12\r\n" );
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document.write( "Edwin

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