document.write( "Question 1209507: Recall that there are 4 suits - spades, hearts, clubs, and diamonds - in a standard deck of playing cards. Suppose you play a game in which you draw a card, record the suit, replace it, shuffle, and repeat until you have observed 10 cards. Define X = numbers of spades observed.\r
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Algebra.Com's Answer #849019 by ikleyn(52798) $\"\"$ $\"About$

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\n" ); document.write( "Recall that there are 4 suits - spades, hearts, clubs, and diamonds - in a standard deck
\n" ); document.write( "of playing cards. Suppose you play a game in which you draw a card, record the suit,
\n" ); document.write( "replace it, shuffle, and repeat until you have observed 10 cards. Define X = numbers of spades observed.\r
\n" ); document.write( "\n" ); document.write( "Find the probability of observing 3 or fewer spades in this game. Round to three decimal places.
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document.write( "In this game, there are 4 suits, and every suit has the same number of card, 13,\r\n" );
document.write( "so the total number of cards in a standard deck is 4*13 = 52.\r\n" );
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document.write( "At each step, this game returns us to the same initial condition, due to replacing and \r\n" );
document.write( "shuffling the cards.\r\n" );
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document.write( "Therefore, this game is a typical binomial experiment.  At each trial, we have a spade \r\n" );
document.write( "with the probability 1/4 = 0.25  or any other suit card with the probability 3/4 = 0.75.\r\n" );
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document.write( "So, they want you calculate the probability of having success (having a spade) 3 or fever \r\n" );
document.write( "times of 10 trials.\r\n" );
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document.write( "Use the standard formula for the binomial probability\r\n" );
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document.write( "    P(X <= 3) = P(0) + P(1) + P(2) + P(3)\r\n" );
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document.write( "where  P(k) = ,   are binomial coefficients   = .\r\n" );
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document.write( "Now calculate using a calculator\r\n" );
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document.write( "     P(0) =  =  = 0.056313515;\r\n" );
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document.write( "     P(1) =  =   = 0.187711716;\r\n" );
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document.write( "     P(2) =  =   = 0.281567574;\r\n" );
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document.write( "     P(3) =  =  = 0.250282288.\r\n" );
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document.write( "Finally, add the number and get \r\n" );
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document.write( "    P(X <= 3) = 0.056313515 + 0.187711716 + 0.281567574 + 0.250282288 = 0.775875092.\r\n" );
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document.write( "Round it to 3 decimals.  So,  P(X <= 3) = 0.776  (rounded).    ANSWER\r\n" );
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document.write( "You may check this result by using a convenient online binomial calculator at\r\n" );
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document.write( "https://stattrek.com/online-calculator/binomial.aspx\r\n" );
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