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Probability and combinatorics

Recent questions in Probability and combinatorics

qtbabe9876a9 2022-05-19

My idea is that by finding the combination of arranging the 10 bears into the sample, this can be used to calculate the probability. For example, this would be $_{10} C_{5} = 30240$ . However, how would I calculate the number of arrangements of the four bears that will go into the sample? I considered $_{5} C_{4}$ , yet this would be illogical as it only includes the number of arrangements of the four bears in the sample, but does not consider the arrangements in the sample not selected. Is my working correct?

adocidasiaqxm 2022-05-17

Probability of getting a sum of 14 when rolling 11 dice.
I assume that in P(A) = m/n, n is 6^11, but I don't know how to calculate the number of ways to get a sum of 14 when rolling 11 dice.

mars6svhym 2022-05-16

Factorial is defined as
$n! = n (n - 1) (n - 2) \dots 1$
But why mathematicians named this thing as factorial?

Brody Collins 2022-05-15

How does the Glivenko-Cantelli theorem improve the stochastic convergence of the empirical distribution $F_{n} (x)$ ?
Let $X_{i}$ be iid random variables with empirical cumulative distribution function $F_{n} (x)$ and CDF $F (x)$ . From the central limit theorem and the strong law of large numbers, we know that $F_{n} \overset{d / a . s .}{\to} F$ . The Glivenko-Cantelli theorem states that $sup_{x \in R} | F_{n} (x) - F (x) | \to 0$ almost surely. How does it impact improvements for these two types of convergence (by itself or maybe by other theorems that are implied)?

rynosluv101wopds 2022-05-15

Let's say that I have two ordered sets of numbers ${1, 2}$ and ${3, 4}$ . I'm trying to figure out the number of possible ways to combine these two sets into one without breaking the ordering of the two sets.
So for instance, ${1, 2, 3, 4}$ , ${3, 4, 1, 2}$ , and ${1, 3, 2, 4}$ are valid combinations, but ${2, 1, 4, 3}$ isn't. How do I figure out the number of valid combinations? This feels like something I should remember from college, but I'm drawing a blank. It feels somewhere in between a combination and a permutation.

Sappeycuii 2022-05-14

given a multiset (a set with repetitions allowed) of $2 n + 1$ real numbers, say $S = {r_{1}, \dots, r_{2 n + 1}}$ .
These numbers are such that for every $k$ , the multiset $S - {r_{k}}$ can be split into two multisets of size $n$ each, such that the sum of numbers in one multiset is same as the sum of numbers in the other.
Show that all the numbers must be equal.( i.e. $r_{i} = r_{j}$ )

Oberhangaps5z 2022-05-13

A large white cube is painted red, and then cut into 27 identical smaller cubes. These smaller cubes are shuffled randomly.
A blind man (who also cannot feel the paint) reassembles the small cubes into a large one. What is the probability that the outside of this large cube is completely red?

Iyana Macdonald 2022-05-12

We have 3 that qualify as best three, say BBB, and 2 as bad say OO.
Thus, the total outcomes can be calculated as follows: 0 bad : This will be 1 as only one way to choose the best three 1 bad : This will be 3 ways. 'OBB' is one example and can be arranged in 3 ways. 2 bad : 3 ways.
Thus, the total outcomes according to me should be 7, not 10. What am I doing wrong?

Oberhangaps5z 2022-05-12

Consider a random arrangement of 20 boys and 16 girls in a line. Let X be the number of boys with girls on both sides. Let Y be the number of girls with boys on both sides. Find $E (X + Y)$
$E (\cdot)$ denotes expectation. For a discrete random variable Z that takes the values $z_{1}, z_{2}, \dots, z_{n}$ with the corresponding probabilities $p_{1}, p_{2}, \dots, p_{n}$ the expectation of Z is defined as $E (Z) = \sum_{i = 1}^{n} z_{i} p_{i}$

Jay Barrett 2022-04-10

Number of ways to stack $N$ cubes.
Once you get a particular stack of $4$ cubes let's say, the order of the cubes doesn't matter, just which sides are being shown does.
I'm generally finding $2$ numbers:
1) $6 * 24^{n - 1}$
and
2) $3 * 24^{n - 1}$
Which is the correct answer, and why?

「ʟᴜᴍɪᴇʀᴇ 」2022-04-02

here are 6 girls and 8 boys at a birthday party. In how many ways can they form six dancing boy-girl couples?

A drug for the relief of asthma can be purchased from 5 different manufacturers in liquid, tablet, or capsule form, all of which come in regular and extra strength. How many different ways can a doctor prescribe the drug for a patient suffering from asthma?

There are 30 students in a class of Mr. Obniarillo. 18 are boys and 12 are girls. What is the probability of
choosing a random student?

rasigapr6 2022-03-06

Write v as a linear combination of $u_{1}, U_{2}$ , and $u_{3}$ , if possible. (Enteryour answer in terms of $u_{1}, U_{2}$ , and $u_{3}$ . If not possible enter IMPOSSIBLE.)
$v = (- 1, 7, 2), u_{1} (3, 2, 8), u_{2} (2, - 3, 1), u_{3} (- 2, 3, - 1)$
v=_____

A statistic instructor ( with at least 20 years of teaching experience with the same course) has established that 10 percent of all students who take his course receive a failing grade. If 10 students have enrolled in his course for the next semester, the probability that more than one of these students will fail is

Is there more variation in "Hard physical activity" or in "Very hard physical activity"?

A population of 19 scores has a mean of 8. One score of X = 6 is removed from the population. What is the value of the new mean?

Alaina Ortiz 2022-02-12

Find the mean, median, and mode for the set of numbers. If necessary, round the mean to one decimal place. 277, 583, 118, 333, 548, 246, 612, 298

Layne Massey 2022-02-11

A bowl has 4 green marbles 3 red marbles and 2yellow marbles what is the probability that you are going to select a red marble and a yellow marble. You replace the marble before another marble is selected

taoigas 2022-01-25

How do you simplify $P_{5}^{2}$ ?

Even though it is always placed into the field of Precalculus, probability and combinatorics problems are more related to analytics and statistical information that many students implement as they are working with the accuracy of their research data. It is one of the reasons why finding good probability combinatorics examples is both complex and not. It is good to have a look through probability combinatorics questions as these will provide you with the answers and help understand how equations work and why possible outcomes are vital for understanding how the calculation takes place.

Precalculus

Combinatorics: Examples, Equations, and Practice Problems