<PRE><B>Question 1158491</B>
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original response deleted... I&#39;m looking at this further....&lt;br&gt;
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In my earlier attempt at solving this, I tried to simplify the problem by treating all the percent increases and decreases as being relative to the original price.  That way, for instance,  increases of 3% and 3% would exactly balance decreases of 1%, 2%, and 3%.&lt;br&gt;
However, that is probably not how the problem was intended.&lt;br&gt;
If the percentage increases or decreases are treated as multipliers, there is NO permutation of 5 of the 6 possible percentage changes that results in an ending price exactly equal to the starting price.&lt;br&gt;
So I looked at the problem again....&lt;br&gt;
Certainly the other tutor missed the point of the problem, finding the percent increase or decrease if each of the changes is applied once.&lt;br&gt;
It appears to me that a purely analytic solution would be extremely tedious, making it necessary to examine each permutation of 5 of the 6 percentage changes.&lt;br&gt;
So I built an excel spreadsheet with all 6^5=7776 permutations of 5 of the 6 and identified the ones that produced a product greater than 1.&lt;br&gt;
ANSWER: 3588 of the 7776 permutations of 5 of the 6 percentage changes produce a product greater than 1.  Therefore, the probability that the stock price will be above the original $10 after 5 days is 3588/7776, or about 46.142%.&lt;br&gt;
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In response to the reader&#39;s question, here is how to make the spreadsheet.&lt;br&gt;
The 6 numbers you are using repeatedly are 1.03, 1.02, 1.01, 0.99, 0.98, and 0.97.&lt;br&gt;
There will be 6^5 = 7776 rows in the table.&lt;br&gt;
In column A there will be 6^4=1296 entries of 1.03, followed by 1296 entries of 1.02, followed by... etc.&lt;br&gt;
In the first 6^4=1296 entries in column B, there will be 6^3=216 entries of 1.03, followed by 216 entries of 1.02, followed by ... etc.&lt;br&gt;
Then you can copy those first 1296 entries in column B 5 more times to complete column B.&lt;br&gt;
So you can see building the spreadsheet is not really as big a task as it seems, because you can do a huge amount of copying and pasting.&lt;br&gt;
Similarly, in column C, you have repeated blocks of 36 entries of 1.03, followed by 36 entries of 1.02, ... etc.&lt;br&gt;
And in column D the repeated pattern is blocks of 6 of each of the 6 numbers.&lt;br&gt;
And finally in column E the 6 numbers repeat in a cycle through the whole 7776 entries.&lt;br&gt;
Now the spreadsheet shows each possible permutation of 5 of the 6 possible numbers exactly once.&lt;br&gt;
Now in column F, make each entry the product of the entries in columns A through E of that row.  Those numbers, being the product of 5 numbers between 0.97 and 1.03, will all be close to 1.  We are of course interested in how many of them are greater than 1.&lt;br&gt;
To count how many of the 7776 entries in this column are greater than 1, there are 2 steps.&lt;br&gt;
(1) in row 1, column G, enter&lt;br&gt;
=IF(F1&gt;1,1,0)&lt;br&gt;
Then copy that formula down through all 7776 rows of column G.&lt;br&gt;
That will put a 1 in column G wherever the value in column F is greater than 1 and a value of 0 wherever that value is less than 1.&lt;br&gt;
Then to count the number of permutations for which the product is greater than 1, simply sum the entries in column G.&lt;br&gt;
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