Box #1 contains five green marbles and fifteen marbles overall; a marble randomly drawn from this box will be green with probability 

Box #2 contains ten green marbles and forty marbles overall; a marble randomly drawn from this box will be green with probability 

, 

so (A) is greater.

Box #1 has eight green marbles out of eighteen marbles total; the probability that a randomly drawn marble is green is .

Box #2 has sixteen red marbles out of thirty-six marbles total; the probability that a randomly drawn marble is green is .

The two probabilities are equal.

The yellow deck has four sixes out of forty-eight cards; the probability of drawing a six at random is .

The gray deck has four sixes out of fifty-two cards; the probability of drawing a six at random is .

Between two fractions with the same numerator, the fraction with the lesser denominator is the greater, so . (A) is greater.

Since there are 30 marbles that are not blue and 15 that are blue, this makes 30 unfavorable outcomes to 15 favorable outcomes. This makes the odds against this favorable outcome, in fraction form:

, 

or 2 to 1 odds.

There are 45 marbles that are neither blue nor green, and twenty-five marbles that are blue or green, so the odds against drawing a blue or green marble are 45 to 25, or

That is, 9 to 5 odds against.

Originally, the box contained twenty-five marbles, ten of which were red, so the probability of drawing a red marble was 

After adding fifteen red marbles, the box contains forty marbles, twenty-five of which are red, so the probability of drawing a red marble is now 

The increase in probability is 

Before removing the twos, there were fifty-two cards, thirteen of which are hearts, making the probability that a randomly-drawn card would be a heart

After removing the twos, there are forty-eight cards, twelve of which are hearts (one heart, the three of hearts, was removed), making the probability that a randomly-drawn card will be a heart

The probability did not change.

We need only compare numbers of cards.

Since no clubs were removed, there are thirteen clubs in the modified deck.

Since one 2, one 3, and one 4 were removed, there are three of each, or nine of these cards total, in the modified deck.

The probability of drawing a club, (A), is greater.

We need only compare numbers of cards.

Since one diamond (the ace of diamonds) was removed, there are twelve diamonds in the modified deck.

Since no jacks, queens, or kings were removed, all four of each (twelve cards total) remain.

This makes the two quantities and, subsequently, the two probabilities, equal.

Box #1 contains  blue marbles out of 30 total. 

Box #2 contains  blue marbles out of 30 total. 

The probability of drawing a blue marble out of Box #1 is the same as that of drawing a blue marble out of Box #2: .