Some STEP probability and mechanics questions

Some STEP probability and mechanics questions from Michael Gibson’s “mini-mock” papers. These questions can be done with relatively short working and without needing to use any great knowledge of formulas.

STEP statistics examples

Worked solutions to:

I/2011/13
I/2015/12
I/2008/12
I/2014/13
I/2016/13
II/2016/13

Download worked solutions here

STEP I 2009 Q.4

STEP II 2009 Q.7

STEP I-2012-13

Question: I choose at random an integer in the range 10000 to 99999, all choices being equally likely. Given that my choice does not contain the digits 0, 6, 7, 8 or 9, show that the expected number of different digits in my choice is 3.3616.

STEP I 2005 Q.2

STEP I-2015-3

A prison consists of a square courtyard of side b bounded by a perimeter wall and a square building of side a placed centrally within the courtyard. The sides of the building are parallel to the perimeter walls.

Guards can stand either at the middle of a perimeter wall or in a corner of the courtyard.

If the guards wish to see as great a length of the perimeter wall as possible, determine which of these positions is preferable.

You should consider separately the cases b < 3a and b > 3a.