Question 964152
1) Cards above a jack : Queen, King, Ace (the set {Queen,King,Ace} )
Cards below a 4 : 2, 3  (the set {2,3} )
Those two sets of cards have no elements in common.
 
a) A card cannot be both above a jack and below a 4,
so the probability that the randomly dealt card is both above a jack and below a 4 is {{{highlight(0)}}} .
 
b) The cards that are either above a jack or below a 4 are the 2, 3, Queen, King, and Ace of every suit
(5 out of the 13 cards in every suit,
and {{{4*5=20}}} out of the {{{4*13=52}}} cards),
so the probability that the randomly dealt card is either above a jack or below a 4 is {{{20/52=highlight(5/13)}}}
 
2) There are 6 possible outcomes for each die, all equally likely, so there are {{{6*6=36}}} possible outcomes, all equally likely.
Some pairs of outcomes will
be indistinguishable, unless we use two dice of different colors.
For example,
a 2 on a red die, and a 1 on a green die
is a different, and equally likely outcome as
a 2 on a green die, and a 1 on a red die, or as 
a 1 on both dice.
So, getting a 1 and a 2 as one of those 2 out of 36 possible outcomes,
with a probability of {{{2/36=1/18}}} ,
while getting 1 on both dice is one of the 36 outcomes, with a {{{1/36}}} probability.
 
For the sum to be less than 4, it must be
{{{1+1=2}}} or {{{2+1=1+2=3}}} ,
with the probabilities described above.
 
For the sum of the numbers to be even, both numbers must be even, or both numbers must be odd. Each die will be odd  half of the time and even half of the time, so if I represent the number on one particular die with capital letters, and the number on the other die with lowercase letters, the possible outcomes can be represented as four equally probable groups:
(ODD,odd) (ODD,even) (EVEN,odd) (EVEN,even) .
Two of those will yield even sums and the other two will yield odd sums.
The probability of an even sum, is {{{1/2}}}, and so is the probability of an odd sum.
 
To get the probability that the sum is even or less than 4,
we must add the probability that the sum is even (1/2),
plus the probability that the sum is {{{+2=2+1=3}}} (1/18).
 
So, the probability that the sum is even or less than 4 is
{{{1/2+1/18=9/18+1/18=10/18=highlight(5/9)}}}