How large a proportion of British and American troops were involved in direct combat with the enemy in World War II? What was the casualty rate among those men?
I believe the "Tooth-to-tail ratio", the ratio of combat personnel to support personnel, is a useful guide for how many troops would have been involved in fighting. This is a slightly dodger proposition today in wars without a front-line (where logistics personal have a very real prospect of being engaged), but seems reasonable for WW2. I happen to be concentrating on American figures rather than British.
The Other End of the Spear: The Tooth-to-Tail Ratio (T3R) in Modern Military Operations by John J. McGrath provides some of the information we need on this. For example it seems that while US command aimed for 65% combat troops, they attained about 40% in the European theatre. Later on, US forces in Korea has almost the same TtT ratio so this seems generally applicable. I will be assuming the theatre TfT ratio this is very close to the Army TfT ratio, but given the size of the army compared to other services this seems reasonable.
Here are Wikiepdia's causalities figures of course. But, the Congress Research Service have some that differentiate by service, so I'll be using these and looking only at the army. There were about 235,000 "battle deaths" and 83,400 other deaths out of about 11.2 million soldiers. Death rate here of 2.8%. Assuming our 40% combat troops ratio applies to the army generally, that gives us about 4.5 million combat personnel. Which results in about a 6% death rate if we attribute all battle deaths and a proportional share of the "other" deaths to these combat personnel.
All these numbers are a bit less than triple that if we include non mortal wounding as well (ie casualties rather than deaths). And I will say I've made more assumptions here than I am happy with, so don't get too excited about these numbers.
But note that it varies from source to source , and maybe a book could give better information's.