document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1117696>Question 1117696</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two events A and B are such that P(A) = 9/16, P(B) = 3/8, and P(A | B) = 1/4.  Find the probability that:<BR>\n" );
document.write( "a. Both events will happen<BR>\n" );
document.write( "b. Only one of the event shows will happen<BR>\n" );
document.write( "C. Neither event will happen  </SPAN>\n" );
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document.write( "If you go with the definition of P(A|B), then<br><BR>\n" );
document.write( "P(A|B) = P(A AND B)/P(B)<br><BR>\n" );
document.write( "One way to look at this definition is that the only part of the sample space you are considering is B, so P(B) is the denominator; and you are concerned with only the part of B that is also A, which makes P(A AND B) the numerator.<br><BR>\n" );
document.write( "For your problem, the calculation is<br><BR>\n" );
document.write( "1/4 = P(A AND B)/(3/8)<BR>\n" );
document.write( "P(A AND B) = (1/4)(3/8) = 3/32<br><BR>\n" );
document.write( "That calculation is in the form<br><BR>\n" );
document.write( "P(A AND B) = P(B)*{P(A|B)<br><BR>\n" );
document.write( "That form of the definition of P(A|B) makes more sense to me personally; it says that to find the probability of both A and B, you start with the probability of B and multiply it by the probability that A is true if B is true.<br><BR>\n" );
document.write( "So we have P(A AND B) = 3/32, which is the first answer we are looking for.<br><BR>\n" );
document.write( "Then P(A only) is P(A) minus P(A AND B) = 9/16-3/32 = 15/32 and P(B only) = 3/8 - 3/32 = 9/32; so the answer to the second question is 15/32+9/32 = 24/32 = 3/4.<br><BR>\n" );
document.write( "The probabilities from the first two parts of the problem cover all the cases where at least one of A or B happens; so the probability that neither event will happen is 1 - 27/32 = 5/32.<br><BR>\n" );
document.write( "Answers: a. 3/32  b. 3/4  c. 5/32 </SPAN>\n" );
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