Probabilities

What is the actual percentage chance of certain events occurring in Warhammer?  As one knows, the game system is centered around six sides dice, each face of which has a 16.67% chance of landing.

Ordinary Space Marines have a ballistic skill of 4.  This means that they hit their target if they roll a 3, 4, 5, or 6 on a six sided dice roll.  As a result their chance of hitting are 4/6 or 66.66%.

Similarly they wound other space marines with a toughness of 4 with bolters of strength 4 with a roll of 4, 5, or 6 on a six sided dice roll.  Their chance of wounding are 3/6 or 50%.

A space marine target wounded by a bolter gets a 3+ armor save, meaning that a roll of a 3, 4, 5, or 6 on a six sided dice roll will save him.  Contrarily a roll of a 1 or 2 will mean an unsaved wound.  The chance of this occurring is 2/6 or 33.33%.

The chance of multiple events all happening is determined by multiplying the percentages of each even occurring separately.

Accordingly the chance of an ordinary space marine wounding another ordinary space marine with a single shot of a bolter is: 66.66% X 50% X 33.33% = 11.11%.

What is the chance of a squad of 10 ordinary space marines firing 10 bolter shots at ordinary space marines to fail to wound any of them?  Well we know that there is an 11.11% chance of registering an unsaved wound.  This means that there's an 88.89% chance of a failed wound.  If we want to see what the chances are of 10 shots all failing we multiply that number 10 times: 88.89% X 88.89% X 88.89% X 88.89% X 88.89% X 88.89% X 88.89% X 88.89% X 88.89% = 31.18%

Contrarily the chance of at least one unsaved wound occurring as a result of 10 shots would be 100% - 31.18% = 68.8%