Suppose that you select 2 cards without replacement from an ordinary deck of playing cards.

a) If the first card that you select is a heart, what is the probability that the second card that you select is
an heart?

b) If the first card that you select is an ace, what is the probability that the second card that you
select is a spade?

a. First card = 13/52, second card = (13-1)/(52-1)

b. If the first card is hearts, clubs or diamonds, spade = 13/(52-1). If the first card is the ace of spades, spades = (13-1)/(52-1)

Carol has one set of 12 trading cards and another set of 15 trading cards. She sorts the cards into 3 equal groups. How many cards are in each group?

To find the probability of events, we need to determine the sample space and the favorable outcomes.

In both cases, we'll assume we have a standard deck of 52 playing cards, where there are 13 hearts and 4 aces.

a) If the first card selected is a heart, then there are 51 cards left in the deck, including 12 hearts. We want to find the probability that the second card drawn is also a heart.

To calculate the probability, we divide the number of favorable outcomes (getting a heart on the second draw) by the total number of possible outcomes.

Favorable outcomes: There are 12 hearts remaining in the deck.
Total number of outcomes: There are 51 cards remaining in the deck.

Therefore, the probability of drawing a heart on the second draw given that the first card is a heart is:
P(heart on second draw | heart on first draw) = 12/51.

b) If the first card selected is an ace, then there are 51 cards left in the deck, including 3 more aces. We want to find the probability that the second card drawn is a spade.

Favorable outcomes: There are 13 spades in the deck, and 3 of them are aces.
Total number of outcomes: There are 51 cards remaining in the deck.

Therefore, the probability of drawing a spade on the second draw given that the first card is an ace is:
P(spade on second draw | ace on first draw) = 3/51.

Note: In both cases, we assume that the cards are not replaced after they're drawn.