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# Probability and combinatorics: FAQ

## What is a Venn diagram and how is it used in probability?

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.

## What is the multiplication rule for probabilities?

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 $\frac{1}{6}$, and the probability of flipping heads on a coin is $\frac{1}{2}$, the probability of rolling a 6 AND flipping heads is $\frac{1}{6}×\frac{1}{2}=\frac{1}{12}$.
When events are not independent, the probability of both A and B happening can be found using the formula:
$P\left(\text{A and B}\right)=P\left(\text{A}\right)\cdot P\left(\text{B}|\text{A}\right)$
Where $P\left(\text{B}|\text{A}\right)$ is the conditional probability of $\text{B}$ happening given that $\text{A}$ has already happened.

## What is the difference between theoretical and empirical probability distributions?

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 $2$ on a fair six-sided die, we would say that the theoretical probability is $\frac{1}{6}$. If we then roll the die $60$ times and get a $2$ in $9$ of those attempts, we can say that the empirical probability is $\frac{9}{60}$.
As we collect more and more data, the empirical probability will converge to the theoretical probability.

## What's the difference between permutations and combinations?

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 $3$ colors from a set of $6$, there are $20$ different combinations if the order doesn't matter, but there are $120$ different permutations if the order does matter.

## What is combinatorics, and how does it help with probability?

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.

## What is a probability distribution?

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 $7$ being the most likely outcome.

## How can probability be used to make fair decisions?

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.

## What are some real-world applications of probability?

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.

## Want to join the conversation?

• "And of course, casinos use probability to ensure that they always win in the long run" Now this is reason why I will never go to casino
• Congrats to everyone who made it all the way through this Precalculus course!!
• You all deserve to be congratulated on making it all the way here on your own! Now, go on and take the final unit test and crush it!
• YEAH!! nice job too all of the people who made it to the last unit! Good luck on all of your future goals!
• how do i find 0.08 times -190 without a calculator
• what he/she said but mathematically it is :
-190x0.08 = -190*8/100
(1 vote)
• A test consists of nine true/false questions.
A student who forgot to study guesses
randomly on every question. What is the
probability that the student answers at least
two questions correctly?

How do find the right formula to use and what is it? Can someone explain?
(1 vote)
• The answer is (number of ways to get at least two right)/(number of total ways to guess answers).

Start with the denominator: there are 9 independent true-false choices. So 2 choices for the first, 4 for the first two, 8 for the first three, and so on. There are 2^9=512 ways to fill in the test.

Now, there are a lot of ways to get two or more correct; we could get exactly two right, exactly three right, exactly four right, and so on. But there are comparatively few ways to NOT get two or more right. We could get none right (there's one way to do that) or we could get exactly one right (there are 9 ways to do that; any question could be the one we get right).

So in total, there are 10 ways to NOT get 2 or more correct. And since there are 512 ways to fill in the test total, there must be 512-10=502 ways to get or more right.

So the probability is 502/512=251/256, about 98%.