document.write( "Question 1074302: Hi there, I need some help with this problem \r
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\n" ); document.write( "\n" ); document.write( "In the dice game of craps two dice are tossed and the sum of the two numbers on the dice is noted. In a modified version of the game a player tosses two four-sided dice. If the sum is less than four the player wins. If the sum is six or more then the player loses. If the sum is anything else, then the player tosses the two dice again and if the sum matches that of their previous sum they win, otherwise they lose.
\n" ); document.write( "i. Let A be the event that a player wins on their first toss of the dice, and let B be the event that a player loses on their first toss of the dice.
\n" ); document.write( "• Find P(A) and P(B)?
\n" ); document.write( "• Are events A and B mutually exclusive? Please justify your answer.
\n" ); document.write( "ii. If a player got a sum of four on their first throw what is the probability that they win?
\n" ); document.write( "iii. Find the probability that a player wins the game.\r
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Algebra.Com's Answer #689004 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Here are the 16 possible throws of 2 4-sided dice:\r\n" );
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document.write( "(1,1) (1,2) (1,3) (1,4) \r\n" );
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document.write( "(2,1) (2,2) (2,3) (2,4)  \r\n" );
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document.write( "(3,1) (3,2) (3,3) (3,4) \r\n" );
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document.write( "(4,1) (4,2) (4,3) (4,4) \r\n" );
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document.write( "So the probability of the sums are \r\n" );
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document.write( "P(2) = P[(1,1)] = 1/16\r\n" );
document.write( "P(3) = P[(1,2)or(2,1)] = 2/16 = 1/8\r\n" );
document.write( "P(4) = P[(1,3)or(2,2)or(3,1)] = 3/16\r\n" );
document.write( "P(5) = P[(1,4)or(2,3)or(3,2)or(4,1)] = 4/16 = 1/4 \r\n" );
document.write( "P(6) = P[(2,4)or(3,3)or(4,2)] = 3/16\r\n" );
document.write( "P(7) = P[(3,4)or(4,3)] = 2/16 = 1/8\r\n" );
document.write( "P(8) = P((4,4)] = 1/16\r\n" );
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document.write( "P(A) = P(win on first throw) = \r\n" );
document.write( "P( < 4) = P(2)+P(3) = 1/16+2/16 = 3/16\r\n" );
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document.write( "P(B) = P(loses on first throw) =\r\n" );
document.write( "P( ≥ 6) P(6)+P(7)+P(8) = 3/16+2/16+1/16 = 6/16 = 3/8\r\n" );
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\n" ); document.write( "• Are events A and B mutually exclusive? Please justify your answer.
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document.write( "Yes, of course, because you can't both win and lose both on the \r\n" );
document.write( "first throw.  Think of the word \"exclusive\" as synonymous with\r\n" );
document.write( "\"excluding\". Winning on the first throw EXCLUDES losing on the\r\n" );
document.write( "first thown, Losing on the first throw EXCLUDES winning on the\r\n" );
document.write( "first throw.  So the events A,B EXCLUDE each other, so they\r\n" );
document.write( "mutually EXCLUDE each other.  So they are MUTUALLY EXCLUDING or\r\n" );
document.write( "MUTUALLY EXCLUSIVE.  \r\n" );
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\n" ); document.write( " ii. If a player got a sum of four on their first throw what is the probability that they win?
P(win|4 on 1st throw) = P(4 on 2nd throw) = 3/16\r\n" );
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\n" ); document.write( "iii. Find the probability that a player wins the game.
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document.write( "To win, one must either\r\n" );
document.write( "1. throw less than 4\r\n" );
document.write( "OR\r\n" );
document.write( "2. throw 4 1st AND throw 4 2nd\r\n" );
document.write( "OR\r\n" );
document.write( "3. throw 5 1st AND throw 5 2nd\r\n" );
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document.write( "We remember that OR implies that we add, whereas\r\n" );
document.write( "AND implies that we multiply, so\r\n" );
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document.write( "P(win) = \r\n" );
document.write( "P( < 4 on 1st throw) +\r\n" );
document.write( "P(4 on 1st throw)×P(4 on 2nd throw) +\r\n" );
document.write( "P(5 on 1st throw)×P(5 on 2nd throw) =\r\n" );
document.write( "3/16 + (3/16)×(3/16) + (1/4)×(1/4) =\r\n" );
document.write( "3/16 + 9/256 + 1/16 = \r\n" );
document.write( "48/256 + 9/256 + 16/256 = 73/256   \r\n" );
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document.write( "Edwin
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