Question 964152: 1. A card is dealt from a complete deck of fifty-two playing cards (no jokers). Count an Ace as high (above a King, not below a 2).
a)Find the probability that the randomly dealt card is both above a jack and below a 4.
b)Find the probability that the randomly dealt card is either above a jack or below a 4.
2. A pair of dice is rolled and the sum of the numbers is observed. What is the probability that the sum is even or less than 4?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! 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  .
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  out of the  cards),
so the probability that the randomly dealt card is either above a jack or below a 4 is 
2) There are 6 possible outcomes for each die, all equally likely, so there are  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  ,
while getting 1 on both dice is one of the 36 outcomes, with a  probability.
For the sum to be less than 4, it must be
 or  ,
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  , 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  (1/18).
So, the probability that the sum is even or less than 4 is
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