1. Basic probability is the likelihood of a particular event happening, expressed as a number between 0 (impossible) and 1 (certain). The formula for calculating probability is:
Probability = Number of favorable outcomes / Number of possible outcomes.

Example: What is the probability of rolling a 5 on a fair six-sided die?
Number of favorable outcomes = 1 (rolling a 5)
Number of possible outcomes = 6
Probability = 1/6 = 0.1667

2. The addition rule for disjoint events states that if two events are disjoint (mutually exclusive), the probability of either of them happening is the sum of their individual probabilities. The formula is:
P(A or B) = P(A) + P(B)

Example: What is the probability of rolling either a 2 or a 4 on a fair six-sided die?
P(2 or 4) = P(2) + P(4) = 1/6 + 1/6 = 1/3 = 0.3333

3. The general addition rule states that for any two events A and B, the probability of either event happening is the sum of their individual probabilities minus the probability of both events happening at the same time. The formula is:
P(A or B) = P(A) + P(B) - P(A and B)

Example: What is the probability of drawing a red card or a face card from a standard deck of cards?
P(red or face) = P(red) + P(face) - P(red face) = 26/52 + 12/52 - 6/52 = 32/52 = 0.6154

4. The multiplication rule for probability states that the probability of two independent events both happening is the product of their individual probabilities. The formula is:
P(A and B) = P(A) * P(B)

Example: What is the probability of flipping two heads in a row with a fair coin?
P(head and head) = P(head) * P(head) = 0.5 * 0.5 = 0.25

5. The multiplication rule for a counting problem is used when the events are dependent, and the outcome of the first event affects the outcome of the second event. The formula is similar to the multiplication rule for probability but takes into account the reduced number of options for the second event based on the outcome of the first event.

Example: What is the probability of drawing a red card and then drawing a black card without replacement from a standard deck of cards?
P(red and black) = P(red) * P(black after red) = 26/52 * 26/51 = 0.2451

6. A permutation is an arrangement of objects in a specific order. It is used when the order matters, such as arranging a sequence of events or selecting a president, vice president, and treasurer from a group of candidates. The formula for permutation is:
nPr = n! / (n-r)!

Example: How many ways can you arrange the letters ABCD?
4P4 = 4! / (4-4)! = 4! / 0! = 24

7. A combination is a selection of objects without considering the order. It is used when the order does not matter, such as forming a committee or choosing a group of players for a team. The formula for combination is:
nCr = n! / (r! * (n-r)!)

Example: How many different ways can you choose 2 people out of a group of 5 to form a committee?
5C2 = 5! / (2!*(5-2)!) = 10

8. The process for finding the possible arrangements of the letters in the word FOOTBALL is different from just a basic permutation because of the repeated letters (O). When calculating permutations of a word with repeated letters, you need to adjust for the duplicates to avoid overcounting. In this case, you would need to divide by the factorial of the number of repeated letters (2! for the two O's) to get the correct number of distinct arrangements.

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