Compound Probability

A compound probability is how likely multiple events will happen. For example, to calculate the probability of flipping heads, and then flipping heads again, multiply the probability of flipping heads the first time by the probability of flipping heads the second time: 
first probability    = 0.5
second probability   = 0.5
compound probability = 0.5 × 0.5 = 0.25 = 25%
In the example above, the probability of flipping heads is the same for both flips. This is because they are independent events, meaning the first flip does not influence the second flip. If instead you were to draw colored marbles from a bag, and calculate the probability of drawing a blue marble followed by another blue marble without putting the first marble back, the probabilities would differ, because after the first draw, there are now fewer blue marbles in the bag, making it harder to draw another blue marble.
To calculate the probability of flipping heads or flipping tails with a single flip, add the probability of flipping heads and the probability of flipping tails: 
first probability    = 0.5
second probability   = 0.5
compound probability = 0.5 + 0.5 = 1 = 100%
In the example above, flipping heads and flipping tails are mutually exclusive, meaning there is no overlap between the two events. Either you flip heads, or you flip tails, both cannot happen. If instead you were to calculate the probability of flipping heads on the first flip or on the second flip, you would add the probability of flipping heads the first time and the probability of flipping heads the second time, and then subtract the probability of flipping heads on both flips since that was double counted: 
first probability    = 0.5
second probability   = 0.5
compound probability = 0.5 + 0.5 - (0.5 × 0.5) = 0.75 = 75%

Exercise 1 of 6

What's the probability of rolling a total of 2 with two 6-sided dice? Said another way, what's the probability of rolling a 1 with one die, and then rolling a 1 again?

Exercise 2 of 6

What's the probability of rolling a total of 3 with two 6-sided dice? There are two ways to roll a total of 3. Calculate the probability of each.

Exercise 3 of 6

In the game Pig, players roll a 6-sided die as many times as they like, and add each roll's value to their turn total. However, if they roll a 1, their turn is over, and they lose all of the points from that turn. Players can choose to pass to add the turn total to their grand total. What's the probability of not rolling a 1 three times in a row?

Exercise 4 of 6

In Texas hold'em, players are dealt two cards, and will later combine those cards with community cards to form a poker hand. If your initial two cards are both aces, it's called pocket aces, and is the best possible starting hand. What's the probability of getting pocket aces? After one card is dealt, there are only 51 cards left in the deck.

Exercise 5 of 6

In Texas hold'em, what's the probability of your initial two cards having at least one ace? Don't double count pocket aces.

Exercise 6 of 6

In blackjack, players are dealt two cards, and can ask for more cards one at a time. Card ranks are added together, with face cards having a value of 10, and aces having a value of 1 or 11. The goal is to have a higher total than the dealer without going over 21. If your initial two cards add up to 21, it's called a blackjack, and is the best possible starting hand. What's the probability of getting a blackjack?

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