A probability (確率) expresses how likely an event (something that happens) is, as a number from 0 to 1.
Count all cases
Count matching cases
Matching ÷ all
The probability of drawing a red ball is 3 ÷ 5 = 0.6.
3 / 3
When every outcome is equally likely, a probability is found as the number of cases in which the event occurs divided by the number of all possible cases. The probability that two events that do not affect each other (independent events) both occur is the product of their probabilities. Multiplying each possible value by its probability and adding the results gives the expected value, which shows the value you can expect on average.
AnalogyA 30% chance of rain in a weather forecast is simply the probability 0.3 written as a percentage.
How to rememberThe 率 in 確率 means a rate, so read a probability as a share with the whole as the denominator.
Exam points
ExamWhen trials that do not affect each other are repeated, the probability that every trial meets the condition is the product of the probabilities of each trial. The probability that an outcome never occurs is found by multiplying together, once per trial, the probability that it does not occur in one trial.
ExamThe expected value (期待値) is the sum of each possible value multiplied by its probability. The expected number of times an outcome occurs is the number of trials × the probability of that outcome.
Mix-upAn expected value is not a probability itself but an average weighted by probabilities, so it need not fall between 0 and 1.
A permutation (順列) count is the total number of ways to pick from several distinct items and arrange them in order.
Arrange 2 of 3 colored flags left and right
Left: Red
Right: Blue
Right: Yellow
Left: Blue
Right: Red
Right: Yellow
Left: Yellow
Right: Red
Right: Blue
Each leaves 2 choices for the right, so there are 3 × 2 = 6 ways.
3 / 3
The number of permutations of r items chosen from n is written nPr and is found by multiplying r numbers that fall by 1 each time, as in n × (n−1) × …. The number of ways to arrange all n items is n × (n−1) × … × 1, called the factorial of n (n!). An arrangement in which the same item can be used any number of times is a permutation with repetition, and with m candidates for each of k positions there are m^k ways.
AnalogyIt is like deciding first, second and third place on a podium. The same three people give a different result whenever their ranks change.
How to rememberThe 列 in 順列 means a line, and once things stand in a line there is a front and a back.
Exam points
ExamWhen an arrangement has conditions, first count the positions or groups that carry the condition, then multiply by the number of ways to arrange the rest. Treat two items that must be adjacent as one block, then multiply by the 2 orders inside the block.
ExamIn a permutation with repetition (重複順列), the number of candidates for each position does not shrink, so placing m candidates in k positions gives m multiplied k times, or m^k ways.
Mix-upnPr multiplies r numbers that fall by 1 each time, while a permutation with repetition m^k multiplies the same m k times, so the calculations differ.
A combination (組合せ) count is the total number of ways to choose items from several distinct ones without regard to order.
Arrangements 5 × 4 = 20
Orders of the same 2 people 2! = 2
20 ÷ 2 = 10 ways
Dividing 20 by 2 gives 10 combinations of 2 people from 5.
3 / 3
The number of combinations of r items chosen from n is written nCr and is found by dividing the permutation count nPr by r!, the number of ways to order the r chosen items. Choosing r items is the same as choosing the n − r items left behind, so nCr = nC(n−r) holds.
AnalogyIt is like picking three ingredients for a hot pot. Whatever order you add them in, the same ingredients make the same pot.
How to rememberThe 組 in 組合せ means a team or group, and a group has no order inside it.
Exam points
ExamOne link between two parties corresponds to one combination of 2 items, and the number of pairs from n items is found as n × (n−1) ÷ 2.
ExamWhen n items become n + 1, the number of pairs grows by n, so the larger n is, the faster the count rises.
Mix-upIf roles or an order are assigned after choosing, count with permutations (順列); if only who is chosen matters, count with combinations.
When all outcomes are equally likely, a probability is the number of matching cases divided by the number of all cases, and it lies from 0 to 1.
The probability that independent events both occur is the product of their probabilities, and an expected value is the sum of each value times its probability.
Count a conditional arrangement starting from the part with the condition and multiply, and when the same item can be reused there are m^k ways.
The number of pairs from n items is n × (n−1) ÷ 2, and it grows faster as n gets larger.