You are on the first level of a steel platform, and you must get to the second level immediately.

On your right, there is a ladder that leads to the second level with rotten wood steps. There are 11 steps to the top, and each step fails 8% of the time when stepped on in this weathered condition. If the step breaks, so does your leg that's on it. These steps are too steep to take more than one at a time. You can grab the sides of the ladder with your hands, but cannot pull yourself up without stepping on the ladder's steps.

On your left there is set of stairs that has 15 steps on it which are slightly more weathered. The stairs are at such an incline that you can skip over one of these steps to get to the next. However, the odd numbered steps fail 10.5% of the time, and the even number of steps fail 12% of the time. As you might have guessed, if the steps break, then so does your leg.

What is the best route to take to avoid breaking your leg?

Let me begin by saying I agree with the previous posts, but I thought of something sneaky.

Take the ladder and lay it down over the stairs. 

Assumptions:

The ladder won't slide down.
The 15 stairs do not include the top and bottom landings 
They are standard sized stairs (7" rise and 10.5" run)
The ladder is just long enough to reach the top when just about vertical.
The ladder has 11 rungs evenly spaced and this space is also equal to that at the top and bottom.
You will only break a leg if a ladder rung and the stair under it both break.

The stairs cover a height of 112" and a length of 168" (from the base of the ladder).  By pythagoras each stair extends a diagonal length of 12.619"

Now the ladder will only reach up a few stairs, and the spacing between rungs is 112/12 = 9.333"

By adding up the spacings the stairs under the rungs are as follows.

Rung #  Stair #

1           0         (No risk as the floor is solid)
2           1
3           2
4           2
5           3
6           4
7           5
8           5
9           6
10         7
11         8

The remaining 7 stairs are not covered by rungs.

There are 5 rungs covering even stairs.  Probability both will break = .08*.12 = .0096
There are 5 rungs covering odd stairs.  Probability both will break = .08*.105 = .0084
Of the seven stairs remaining you probably couldn't jump past the ninth, so you would have to take the last 4 odd stairs.  Probability each will break = .105

You want none of these 14 footfalls to result in a broken leg so wee want the product of 1 - each of these probabilities.

.9904^5 * .9916^5 * .895^4 = .5862

A better result :-)

Posted by Jer on 2005-04-26 18:15:49