document.write( "Question 858827: In a family with 5 children, assuming the probability of having a boy or a girl is equally likely, what is the probability of having exactly 4 boys? \n" ); document.write( "
Algebra.Com's Answer #517423 by math-vortex(648)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "Hi there,\r\n" );
document.write( "\r\n" );
document.write( "The Problem:\r\n" );
document.write( "In a family with 5 children, assuming the probability of having a boy or a girl is equally likely, what is the probability of having exactly 4 boys?\r\n" );
document.write( "\r\n" );
document.write( "Solution:\r\n" );
document.write( "This situation can be modeled by the Binomial Distribution because:\r\n" );
document.write( "\r\n" );
document.write( "1: The number of observations n is fixed :: 5 children in the family.\r\n" );
document.write( "2: Each observation is independent :: each child's gender is independent of the others.\r\n" );
document.write( "3: Each observation represents one of two outcomes :: child is a girl or a boy.\r\n" );
document.write( "4: The probability of \"success\" p is the same for each outcome :: probability of having a boy is always the same.\r\n" );
document.write( "\r\n" );
document.write( "The formula for the Binomial Distribution is \r\n" );
document.write( "\r\n" );
document.write( "P(X = k) = [nCk] * [p^k] * [q^(n-k)]\r\n" );
document.write( "\r\n" );
document.write( "What do all these variables mean?\r\n" );
document.write( "\r\n" );
document.write( "For having exactly 4 boys: \r\n" );
document.write( "X = the random variable representing the event \"the child is a boy\".\r\n" );
document.write( "n = the number of trials (number of children) = 5\r\n" );
document.write( "k = the number of successful trials (It's a boy.) = 4\r\n" );
document.write( "n-k = the number of unsuccessful trials (wrong answers) = 5-4 = 1\r\n" );
document.write( "p = the probability of success, \"having a boy\" = 1/2 = 0.5\r\n" );
document.write( "q = the probability of failure, \"having a girl\" = 1/2 = 0.5\r\n" );
document.write( "\r\n" );
document.write( "nCk = the combination \"n choose k,\" how many ways to choose k items from a set of n items = 5C4\r\n" );
document.write( "nCK = n!/(k!(n-k)!) = 5!/(1! 4!)\r\n" );
document.write( "P(X = k) = the probability that with random guessing there will be exactly k boys.\r\n" );
document.write( "\r\n" );
document.write( "Your problem asks for the probability that there will be exactly 4 boys, P(k=4)\r\n" );
document.write( "\r\n" );
document.write( "P(k=4) = [5C4] * [(0.5)^4] * [(0.5)^1]= [5] = 0.15625\r\n" );
document.write( "\r\n" );
document.write( "The probability of having 4 boys in a family of 4 children is 0.15625.\r\n" );
document.write( "\r\n" );
document.write( "If you have questions about this, or you want to check your answer, feel free to email me.\r\n" );
document.write( "\r\n" );
document.write( "Mrs. Figgy\r\n" );
document.write( "math.in.the.vortex@gmail.com\r\n" );
document.write( "
\n" ); document.write( "
\n" );