document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207232>Question 1207232</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A volleyball team is playing in a tournament. The probability that they will win their first match is 60%. The coach noticed that when the team wins a match, the probability that they will win the next match rises to 80%. If they lose a match, the probability that they will win the next match falls to 25%.\r<BR>\n" );
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document.write( "a) Write the initial probability matrix and the transition matrix.\r<BR>\n" );
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document.write( "b) Determine the probability that the team will win its third match. </SPAN>\n" );
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document.write( "Let's for a moment do this without use of Markov notation and matrices. <BR>\n" );
document.write( "The next section will handle that. You can skip directly to that section if you prefer.\r<BR>\n" );
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document.write( "Events<BR>\n" );
document.write( "W1 = win 1st match<BR>\n" );
document.write( "W2 = win 2nd match<BR>\n" );
document.write( "W3 = win 3rd match<BR>\n" );
document.write( "L1 = lose 1st match<BR>\n" );
document.write( "L2 = lose 2nd match<BR>\n" );
document.write( "L3 = lose 3rd match\r<BR>\n" );
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document.write( "For the first match<BR>\n" );
document.write( "P(W1) = 0.60<BR>\n" );
document.write( "P(L1) = 0.40\r<BR>\n" );
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document.write( "If the team wins the first match,<BR>\n" );
document.write( "P(W2 given W1) = 0.80<BR>\n" );
document.write( "P(L2 given W1) = 0.20<BR>\n" );
document.write( "The \"given\" refers to conditional probability.<BR>\n" );
document.write( "You can use a vertical bar to represent the key term \"given\", but I prefer to use the word to avoid potential confusion with the number one or lowercase L.\r<BR>\n" );
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document.write( "If the team loses the first match,<BR>\n" );
document.write( "P(W2 given L1) = 0.25<BR>\n" );
document.write( "P(L2 given L1) = 0.75\r<BR>\n" );
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document.write( "Use the Law of Total Probability to get:<BR>\n" );
document.write( "P(W2) = P(W2 and W1) + P(W2 and L1)<BR>\n" );
document.write( "P(W2) = P(W2 given W1)*P(W1) + P(W2 given L1)*P(L1)<BR>\n" );
document.write( "P(W2) = 0.80*0.60 + 0.25*0.40<BR>\n" );
document.write( "P(W2) = 0.48 + 0.10<BR>\n" );
document.write( "P(W2) = 0.58<BR>\n" );
document.write( "which immediately gives us P(L2) = 1-P(W2) = 1-0.58 = 0.42\r<BR>\n" );
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document.write( "A probability tree diagram might be helpful.<BR>\n" );
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document.write( "The numbers at the very bottom represent multiplications along that specific branch.<BR>\n" );
document.write( "Example: 0.60*0.80 = 0.48 along the left-most pathway<BR>\n" );
document.write( "Notice P(W2) = P(W2 and W1) + P(W2 and L1) = 0.48+0.10 = 0.58\r<BR>\n" );
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document.write( "We have:<BR>\n" );
document.write( "P(W2) = 0.58<BR>\n" );
document.write( "P(L2) = 0.42\r<BR>\n" );
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document.write( "If the team wins the second match,<BR>\n" );
document.write( "P(W3 given W2) = 0.80<BR>\n" );
document.write( "P(L3 given W2) = 0.20\r<BR>\n" );
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document.write( "If the team loses the second match,<BR>\n" );
document.write( "P(W3 given L2) = 0.25<BR>\n" );
document.write( "P(L3 given L2) = 0.75\r<BR>\n" );
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document.write( "Apply the Law of Total Probability once again:<BR>\n" );
document.write( "P(W3) = P(W3 and W2) + P(W3 and L2)<BR>\n" );
document.write( "P(W3) = P(W3 given W2)*P(W2) + P(W3 given L2)*P(L2)<BR>\n" );
document.write( "P(W3) = 0.80*0.58 + 0.25*0.42<BR>\n" );
document.write( "P(W3) = <font color=red>0.569</font> is probability the team wins the 3rd match.\r<BR>\n" );
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document.write( "Once again, a probability tree diagram might help visualize what's going on here.<BR>\n" );
document.write( "This is what the template will look like<BR>\n" );
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document.write( "I'll let the student fill it out.\r<BR>\n" );
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document.write( "Now we'll apply Markov mathematics.\r<BR>\n" );
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document.write( "Your teacher presented you with 3 facts<ul><li>Fact 1: The probability that they will win their first match is 60%</li><li>Fact 2: When the team wins a match, the probability that they will win the next match rises to 80%. </li><li>Fact 3: If they lose a match, the probability that they will win the next match falls to 25%.</li></ul>W = win<BR>\n" );
document.write( "L = lose\r<BR>\n" );
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document.write( "Draw two circles. Label one of them W and the other L.\r<BR>\n" );
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document.write( "From circle W, draw an arrow that will point back to W. It loops around.<BR>\n" );
document.write( "Label this arrow as 0.80<BR>\n" );
document.write( "It represents the idea that if we start with a win, then the probability of another win is 80%. See fact 2 above.<BR>\n" );
document.write( "From circle W draw an arrow that points to L. This arrow is labeled 0.20\r<BR>\n" );
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document.write( "From circle L draw arrows that point to itself and to W; they will be labeled 0.25 and 0.75 respectively. These labels are due to fact 3.\r<BR>\n" );
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document.write( "Here is the arrow diagram<BR>\n" );
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document.write( "Such a diagram is optional, but may be helpful.\r<BR>\n" );
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document.write( "Now we'll build the transition matrix.<BR>\n" );
document.write( "The rows will represent the initial state and the columns the final state. <BR>\n" );
document.write( "I like to think of it as reading from left to right \"start --> finish\". <BR>\n" );
document.write( "We'll start with a certain row and end up at a certain column.\r<BR>\n" );
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document.write( "Here's what the blank template looks like<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >W</TD><TD >L</TD></tr><tr><TD >W</TD><TD ></TD><TD ></TD></tr><tr><TD >L</TD><TD ></TD><TD ></TD></tr>\n" );
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document.write( "And here's what happens when filling out the transition matrix.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >W</TD><TD >L</TD></tr><tr><TD >W</TD><TD >0.8</TD><TD >0.2</TD></tr><tr><TD >L</TD><TD >0.25</TD><TD >0.75</TD></tr>\n" );
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document.write( "Example: row1 --> column2 represents going from a win to a loss, of which has probability 0.20 = 20%\r<BR>\n" );
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document.write( "Label this as matrix T.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T+=+%28matrix%282%2C2%2C0.80%2C0.20%2C0.25%2C0.75%29%29 ALIGN=MIDDLE  ALT=\"T+=+%28matrix%282%2C2%2C0.80%2C0.20%2C0.25%2C0.75%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Adding along any given row results in a sum of 1.\r<BR>\n" );
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document.write( "The initial probability matrix is going to consist of 1 row and 2 columns. <BR>\n" );
document.write( "The 1 row represents the initial state, which doesn't have a win or loss tag to it yet.<BR>\n" );
document.write( "The 2 columns represent the final state after that 1st match either W or L.\r<BR>\n" );
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document.write( "The initial probability matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+%28matrix%281%2C2%2C0.60%2C0.40%29%29 ALIGN=MIDDLE  ALT=\"P+=+%28matrix%281%2C2%2C0.60%2C0.40%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which should be fairly self explanatory. See fact 1.\r<BR>\n" );
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document.write( "Computing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2AT+=+%28matrix%281%2C2%2C0.60%2C0.40%29%29%2A%28matrix%282%2C2%2C0.80%2C0.20%2C0.25%2C0.75%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > will lead to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2AT+=+%28matrix%281%2C2%2C0.58%2C0.42%29%29 ALIGN=MIDDLE  ALT=\"P%2AT+=+%28matrix%281%2C2%2C0.58%2C0.42%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which matches what we got in the first section above (when we arrived at P(W2) = 0.58 and P(L2) = 0.42)<BR>\n" );
document.write( "I'll leave the matrix multiplication scratch work for the student to do.\r<BR>\n" );
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document.write( "Once we determine matrix PT, we will then apply another multiplication with the transition matrix T to compute PT*T = PT^2<BR>\n" );
document.write( "This will determine the probability values for the 3rd match.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2AT%5E2+=+P%2AT%2AT+=+%28matrix%281%2C2%2C0.60%2C0.40%29%29%2A%28matrix%282%2C2%2C0.80%2C0.20%2C0.25%2C0.75%29%29%2A%28matrix%282%2C2%2C0.80%2C0.20%2C0.25%2C0.75%29%29+=%28matrix%281%2C2%2C0.58%2C0.42%29%29%2A%28matrix%282%2C2%2C0.80%2C0.20%2C0.25%2C0.75%29%29+=+%28matrix%281%2C2%2C0.569%2C0.431%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This aligns with what we got in the first section above.\r<BR>\n" );
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document.write( "Answers:\r<BR>\n" );
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document.write( "(a) Initial probability matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+%28matrix%281%2C2%2C0.60%2C0.40%29%29 ALIGN=MIDDLE  ALT=\"P+=+%28matrix%281%2C2%2C0.60%2C0.40%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and transition matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T+=+%28matrix%282%2C2%2C0.80%2C0.20%2C0.25%2C0.75%29%29 ALIGN=MIDDLE  ALT=\"T+=+%28matrix%282%2C2%2C0.80%2C0.20%2C0.25%2C0.75%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "P is a 1x2 matrix while T is a 2x2 matrix.\r<BR>\n" );
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document.write( "(b) The probability the team wins its third match is 0.569 = 56.9%<BR>\n" );
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