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2026-05-20T07:51:51Z
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Ryan Moruzzi: Challenge Problem Solutions
2015-05-02T17:32:51Z
<p>Challenge Problem Solutions</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:32, 2 May 2015</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In order for no cats to collide, all four cats must walk in the same direction. Thus, all four cats must walk clockwise OR all four cats must walk counterclockwise. This means <math>P(no cats collide) = P(<<del class="diffchange diffchange-inline">text</del>>all cats go clockwise<<del class="diffchange diffchange-inline">/text</del>>) + P(all cats go counterclockwise)</math>. </div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In order for no cats to collide, all four cats must walk in the same direction. Thus, all four cats must walk clockwise OR all four cats must walk counterclockwise. This means <math>P(<ins class="diffchange diffchange-inline"></math></ins>no cats collide<ins class="diffchange diffchange-inline"><math></ins>) = P(<<ins class="diffchange diffchange-inline">/math</ins>>all cats go clockwise<<ins class="diffchange diffchange-inline">math</ins>>) + P(<ins class="diffchange diffchange-inline"></math></ins>all cats go counterclockwise<ins class="diffchange diffchange-inline"><math></ins>)</math>. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The probability a cat chooses clockwise or counterclockwise is <math>1/2</math>. So the probability all four cats choose clockwise or counterclockwise is <math>(1/2)*(1/2)*(1/2)*(1/2)</math>. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The probability a cat chooses clockwise or counterclockwise is <math>1/2</math>. So the probability all four cats choose clockwise or counterclockwise is <math>(1/2)*(1/2)*(1/2)*(1/2)</math>. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Therefore, <math>P(no cats collide) = (1/2)*(1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2)*(1/2) = 1/16 + 1/16 = 1/8.</math></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Therefore, <math>P(<ins class="diffchange diffchange-inline"></math></ins>no cats collide<ins class="diffchange diffchange-inline"><math></ins>) = (1/2)*(1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2)*(1/2) = 1/16 + 1/16 = 1/8.</math></div></td></tr>
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Ryan Moruzzi
https://gradwiki.math.ucr.edu/index.php?title=Answer&diff=654&oldid=prev
Ryan Moruzzi: /* Challenge Problem 2 Solution */
2015-05-02T17:31:10Z
<p><span dir="auto"><span class="autocomment">Challenge Problem 2 Solution</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:31, 2 May 2015</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l2" >Line 2:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In order for no cats to collide, all four cats must walk in the same direction. Thus, all four cats must walk clockwise OR all four cats must walk counterclockwise. This means <math>P(no cats collide) = P(all cats go clockwise) + P(all cats go counterclockwise)</math>. </div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In order for no cats to collide, all four cats must walk in the same direction. Thus, all four cats must walk clockwise OR all four cats must walk counterclockwise. This means <math>P(no cats collide) = P(<ins class="diffchange diffchange-inline"><text></ins>all cats go clockwise<ins class="diffchange diffchange-inline"></text></ins>) + P(all cats go counterclockwise)</math>. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The probability a cat chooses clockwise or counterclockwise is <math>1/2</math>. So the probability all four cats choose clockwise or counterclockwise is <math>(1/2)*(1/2)*(1/2)*(1/2)</math>. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The probability a cat chooses clockwise or counterclockwise is <math>1/2</math>. So the probability all four cats choose clockwise or counterclockwise is <math>(1/2)*(1/2)*(1/2)*(1/2)</math>. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore, <math>P(no cats collide) = (1/2)*(1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2)*(1/2) = 1/16 + 1/16 = 1/8.</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Therefore, <math>P(no cats collide) = (1/2)*(1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2)*(1/2) = 1/16 + 1/16 = 1/8.</math></div></td></tr>
</table>
Ryan Moruzzi
https://gradwiki.math.ucr.edu/index.php?title=Answer&diff=653&oldid=prev
Ryan Moruzzi: Challenge Problem Answers
2015-05-02T17:29:58Z
<p>Challenge Problem Answers</p>
<p><b>New page</b></p><div>==Challenge Problem 2 Solution==<br />
<br />
<br />
In order for no cats to collide, all four cats must walk in the same direction. Thus, all four cats must walk clockwise OR all four cats must walk counterclockwise. This means <math>P(no cats collide) = P(all cats go clockwise) + P(all cats go counterclockwise)</math>. <br />
<br />
The probability a cat chooses clockwise or counterclockwise is <math>1/2</math>. So the probability all four cats choose clockwise or counterclockwise is <math>(1/2)*(1/2)*(1/2)*(1/2)</math>. <br />
<br />
Therefore, <math>P(no cats collide) = (1/2)*(1/2)*(1/2)*(1/2) + (1/2)*(1/2)*(1/2)*(1/2) = 1/16 + 1/16 = 1/8.</math></div>
Ryan Moruzzi