Probability is a very broad topic. Many different questions can be asked

Important info:

1. A deck of cards has 52 cards, with 13 of each symbol and 4 of each number/face. The face cards are jack, queen, king, and ace, and jokers are never part of these questions. Also, there are 16 face cards, as 4 of a card for each symbol (hearts, clubs, spades, diamonds) * 4 different types of face cards.
2. Independent events are two events totally independent of each other, such as the roll of a die and the toss of a coin. Dependent events are events that depend on each other

Mutually Exclusive are different from independent because they deal with only thing, such as only cards or only coins, instead of both cards and coins. Mutually exclusive means two events cannot occur at the same time.

Example 1

Compute the probability of getting a “blackjack” (two cards totaling 21 points) from a standard deck of cards, given that an ace counts as 11, a face card counts as 10, and all others count as their numerical values.

First, calculate the denominator, which is the number of total possibilities from the deck of cards. The first card has 52 possibilities (52 cards in a deck) and the second card picked has 51 possibilities since one of the cards was already picked. Thus, the denominator is 52*51=2652.

Now, calculate the number of possibilities for 21. Since a face card and an ace are the ONLY possibilities to get 21, multiply the number of face cards by the number of aces.

$$ (4*4)*4 = 64 $$

HOWEVER, the ace can be picked first and then a face card of vice versa, so multiply this by 2 to get 128.

$$ \frac{128}{2652}=\frac{32}{663} $$

Example 2

Consider a bag of five marbles, 3 red and 2 white. You draw marbles from the bag until you draw the last white marble, at which point you stop. What is the probability you stop drawing after the 2nd marble?

This means that white is drawn on both the first and second draws. So, the first probability of drawing a white marble on the first draw is

$$ \frac{2\space\text{white}}{2+3\space\text{marbles}}=\frac{2}{5} $$