WORDS WITH LOGICAL CONTENT IN PROBABILITY PROBLEMS

 The union of two sets, A and B, written A B, is the set of all objects which are either a member of the the set A or a member of the set B. The intersection of two sets, A and B, written A B, is the set of all objects which are both a member of the the set A and a member of the set B. The complement of a set A, written A', is the set of all objects which are not a member of the the set A. (Conditional probability) The probability of the event A, given that the event B has occured, written P(A | B), is equal to P(A B) divided by P(B)

The RED WORDS (also underlined) above have logical content, revealing the stucture of a set, or its relationship to another set. These RED WORDS have many synonyms in English, and the words are not always used with logical content. Thus, in [2] above, the word "and" occurs twice; the first "and" has little logical content, merely noting two sets; the second "and" has logical content, linking two properties (or qualities) of points in A B.

EXAMPLES OF LOGICAL CONTENT IN PROBLEMS
Red words (also underlined) have logical content.

[11]
 Two people are selected randomly from a group of 12 Republicans and 10 Democrats. [a] Find the probability that both are Democrats. (See [2] above) [b] Find the probability the one is a Republican and one is a Democrat. (See [2] above) [c] Find the probability that neither is a Republican. (See [3] above)

[12]
 Two good bulbs and seven bad bulbs are mixed in a box. Two bulbs are withdrawn. [a] Find the probability that at least one bulb is good. (See [1] above) [b] Find the probability the one bulb is good and the other is not. (See [2] above) [c] Find the probability that neither is a bad bulb. (See [3] above)

Note that the word "not" above is not used logically, but is used only to identify a bulb type. This "not" could be replaced with "bad".

[13]
 Three people are selected from a group of seven men and 5 women. [a] Find the probability that the first two are women and the third is a man. (See [2] above) [b] Find the probability that the first two are women if the third is a man. (See [4] above) [c] Find the probability that the first two are women given the third is a man. (Same as [b] above)

[14]
 Soft drinks in a cooler are 55% Pepsi and 45% coke. 40% of the Pepsi are diet, and 30% of the coke are diet. A drink is randomly selected and found to be a diet drink.. (See [4] above for all red underlined words) [a] Find the probability that the drink is a Pepsi. [b] Find the probability that the drink is a not a Pepsi. (See [3] above)

Note that the entire red underlined sentence above could have been re-written as "GIVEN THAT a randomly selected drink was found to be a diet drink.", which makes this look more like [4] above.