Probability
Probability equals the number of favourable outcomes divided by the total number of possible outcomes. It lies between 0 and 1, with 1 for a certain event and 0 for an impossible one. The probability of the complement is 1 minus the probability of the event. For any two events, the probability of A or B equals P of A plus P of B minus P of both. For independent events, the probability of both equals the product. For mutually exclusive events, the probability of either is the sum.
Standard Sets
A coin has 2 outcomes and n coins have 2 to the n. A die has 6 outcomes and two dice have 36. On two dice, a sum of 7 can happen in 6 ways, a sum of 8 in 5 ways, and sums of 2 or 12 in 1 way each. A deck has 52 cards in 4 suits of 13, with 26 red, 26 black, 12 face cards and 4 aces.
Permutations
The number of arrangements of r items chosen from n distinct items is n factorial over n minus r factorial. Arranging all n distinct items gives n factorial. Factorials: 3 is 6, 4 is 24, 5 is 120 and 6 is 720. If p items are alike and q items are alike, arrangements are n factorial over p factorial times q factorial. Seating n people around a circular table gives n minus 1 factorial.
Combinations
The number of ways to choose r items from n, when order does not matter, is n factorial over r factorial times n minus r factorial. It equals the choice of n minus r. Permutations equal combinations times r factorial. For committees, choose each group separately and multiply.
Shortcuts
Arranging means permutation and selecting means combination. For "at least one", use one minus the probability of none. Choosing two from n gives n times n minus 1 over 2.
Worked Examples
Example 1: A coin is tossed three times. What is the probability of exactly two heads?
Solution: There are 8 outcomes and 3 of them give exactly two heads, so the probability is 3 over 8.
Example 2: Find the probability that two dice sum to 8.
Solution: There are 5 favourable outcomes out of 36, so the answer is 5 over 36.
Example 3: In how many ways can the letters of LEVEL be arranged?
Solution: 5 factorial over 2 factorial times 2 factorial equals 30.
Quick Check
Question 1: What is n factorial over r factorial times n minus r factorial called?
Answer: Combination, n choose r.
Question 2: How many ways can 6 people sit around a circular table?
Answer: 5 factorial, which is 120.
Question 3: What is the probability of the complement?
Answer: One minus the probability of the event.