Probability and combinatorics: FAQ

A Venn diagram is a graphical representation of the relationships between different sets of items. In probability, we often use Venn diagrams to visualize how different events might overlap or be mutually exclusive. This can help us figure out how to use the addition rule to calculate the probability of one event or the other happening.

The multiplication rule states that the probability of two independent events both happening is equal to the product of their individual probabilities. For instance, if the probability of rolling a 6 on a six-sided die is $\dfrac{1}{6}$start fraction, 1, divided by, 6, end fraction, and the probability of flipping heads on a coin is $\dfrac{1}{2}$start fraction, 1, divided by, 2, end fraction, the probability of rolling a 6 AND flipping heads is $\dfrac{1}{6} \times \dfrac{1}{2} = \dfrac{1}{12}$start fraction, 1, divided by, 6, end fraction, times, start fraction, 1, divided by, 2, end fraction, equals, start fraction, 1, divided by, 12, end fraction.

When events are not independent, the probability of both A and B happening can be found using the formula:

Where $P(\text{B}|\text{A})$P, left parenthesis, start text, B, end text, vertical bar, start text, A, end text, right parenthesis is the conditional probability of $\text{B}$start text, B, end text happening given that $\text{A}$start text, A, end text has already happened.

Theoretical probability distributions are based on mathematical calculations, while empirical probability distributions are based on observed data. For example, if we calculate the probability of rolling a $2$2 on a fair six-sided die, we would say that the theoretical probability is $\dfrac{1}{6}$start fraction, 1, divided by, 6, end fraction. If we then roll the die $60$60 times and get a $2$2 in $9$9 of those attempts, we can say that the empirical probability is $\dfrac{9}{60}$start fraction, 9, divided by, 60, end fraction.

As we collect more and more data, the empirical probability will converge to the theoretical probability.

Both permutations and combinations refer to the ways we can select and arrange objects from a larger set. The difference is that permutations take order into account, while combinations do not. For example, if we want to choose $3$3 colors from a set of $6$6, there are $20$20 different combinations if the order doesn't matter, but there are $120$120 different permutations if the order does matter.

Combinatorics is the branch of mathematics that deals with counting possible outcomes or arrangements, and it can be very useful in probability. For example, if we want to calculate the probability of getting a certain poker hand, we need to know how many different ways that hand can be formed, as well as the total number of possible poker hands. Combinatorics can help us with both of these counts.

A probability distribution describes the likelihood of all possible outcomes for a given event. For example, if we roll a six-sided die, we might say that the probability distribution is uniform, meaning that each side has an equal chance of coming up. On the other hand, if we roll two six-sided dice, the probability distribution is different, with the sum of $7$7 being the most likely outcome.

There are many ways in which probability can be used to make decisions that are fair to everyone involved. For example, if two people want to split a piece of cake but can't decide who gets the first piece, they could flip a coin to randomly decide. Another example might be a teacher who wants to call on students in class in a way that doesn't favor any one student, so they use a random number generator to choose who to call on next.

Probability is used in all sorts of ways in the real world! For example, businesses often use probability to make decisions about what products to develop or how to allocate resources. Sports analysts use probability to predict the outcome of games or tournaments. Insurance companies use probability to calculate the likelihood of different types of accidents or disasters, which helps them set rates and premiums. And of course, casinos use probability to ensure that they always win in the long run.