In each test, the probability of occurrence of event a is p, and the probability of occurrence of odd times of event a in n independent tests is calculated

In each test, the probability of occurrence of event a is p, and the probability of occurrence of odd times of event a in n independent tests is calculated

Answer: [1 - (1-2p) ^ 2] / 2
In n independent repeated trials
The probability of event a occurring once is C (n, 1) * (1-p) ^ (n-1) * P ^ 1;
The probability of event a occurring three times is C (n, 3) * (1-p) ^ (n-3) * P ^ 3;
The probability of event a occurring five times is C (n, 5) * (1-p) ^ (N-5) * P ^ 5;
…………………………………………………………………
They are all even terms in the expansion of binomial [(1-p) + P] ^ n
The sum of them is the sum of even terms in the expansion of
On the expansion of binomial [(1-p) + P] ^ n
And the expansion of binomial [(1-p) - P] ^ n... (2)
All the odd items of them correspond to the same, and all the even items correspond to the opposite
So [(1) - (2)] / 2 is as follows:
The probability is {[(1-p) + P] ^ n - [(1-p) - P] ^ n} / 2
＝ [ 1 - (1-2p)^n ] / 2
If you are satisfied, please click [satisfied] in the upper right corner~

In Bernoulli's probability model, the probability of event a occurring in each experiment P (a) = P, then the probability of occurrence of K times in n independent experiments is (where p + q = 1

P=C(n,k)p^k·q^(n-k) (k=0,1,2,3···,n）
C (x, y) x is a subscript and Y is a superscript

In the following formula of Bernoulli test, the alphabet PN (k) = cnkpkqn-k (0 ≤ K ≤ n)
There is no sign between them. Does it mean multiplication? What does the following () in brackets mean? Does it mean multiplication?
And this: the probability of event a occurring at most m times: m p {0 ≤ ξ ≤ m} = ∑ cnkpkqn-k. I only have junior high school education. What's the relationship between the front m and the back formula? Why is it so far apart

Bernoulli experiment is to repeat n experiments, the probability of success K (from 0 to N, the number of possible success) times. Where p is the probability of success (or can be understood as the probability of some phenomenon), q is the probability of failure (P = 1-Q), and CNK is to find K times (permutation and combination) success from n experiments. The whole test is multiplied

In the n-fold Bernoulli type, let the probability of occurrence of event a in each test be p, then the probability of occurrence of event a exactly k times is?

Let CNK = t (where n is the subscript of C and K is the superscript of C)
Probability s = t * P (to the K power of) * (1-p) (to the n-k power of)
CNK = t (where n is the subscript of C and K is the superscript of C) is the combined expression

Third order magic cube layer formula (letter)

　　f u fi u f u2 fi
Before I start, I would like to suggest three learning methods. One is to watch pictures, animations and videos. I recommend watching the videos first and then reading the text course. Two is to think more by yourself, to explore more by yourself, to let yourself lead and not to rely on the course. Three is to think simply and not to complicate the problem. All the steps are very simple, Let's take a closer look at the text instructions in the tutorial. At this time, we should take a look with problems. The more detailed we can see, the better. In addition, hundreds of thousands of people have successfully restored according to this tutorial. There should be no mistakes in it. Please trust it. That's all. Let's start challenging the Rubik's cube
If you want to learn from our Rubik's cube video tutorial, you should first look at the introduction of the video tutorial
(step 1) make a cross on the first side to form the following appearance: (you can also see the video explanation of step 1)
Note that for a good cross, it must be as shown in the figure above. The edges and center of each side are the same color. There are many ways to make this step, and I suggest you play it freely. If it's really difficult, I'll provide a perfect way, that is, change the edge color block with blue in the middle layer to the bottom, and then adjust the side color, Let me give you an example. You should understand that for position B on the left, you only need the following three steps, magic cube
The animation will play automatically. You can also watch it step by step with the one on the right of the playback bar. The one on the left will return to the original state
　　F D R2
In the initial state, turn the blue and yellow block to the bottom, rotate the bottom, and turn the yellow block to the side
I'm clear enough. Position D and position B are exactly the same, which is to change the blue edge block in the middle layer to the bottom, and then adjust the side color, and then turn it up. For positions a and C, you can rotate the plane to position B or D. if the blue side of a certain edge block is on the top or bottom, I believe you will have a solution, If you have difficulty, change the blue color to the bottom. You can rotate freely on the bottom
Sometimes, you will encounter a situation like the one on the left. When the blue and yellow block turns to the bottom, it will affect the red surface that has been aligned. At this time, you need to restore the position of the red surface before the last step. See the following for the specific operation
Rubik's cube station)
　　F D F' R2
In the initial state, turn the blue and yellow block to the bottom, but
This affects the rotation of the red side to the bottom, the recovery of the yellow side to the red side, and the blue yellow block to the top
According to the above, if you repeat 4 edges, you should be able to make a cross
Here I suggest a clearer and faster method for beginners. After the blue edge changes to the bottom, you can turn it up without rushing. You can turn it into the image on the left. Note that the four edges on the bottom can be in any order, which will reduce your difficulty. Here are two examples to broaden your mind,
Finally, just turn them up one by one
Now I'm going to talk about marks. You may have noticed some strange letters in the last table. The meaning of those letters is very simple,
F = front FACEB = back facer = right face
L = left face u = up face d = down face
Take the mark in the above table as an example, f represents the front turning 90 ° clockwise, f 'represents the front turning 90 ° counterclockwise, R2 represents the right turning 180 ° so simple, you understand
In addition, if your Rubik's cube is a Rubik's cube with numbers, or a Rubik's cube with graphics or patterns, then your six side center block has the problem of orientation. You can refer to this page to correct the side center block in this step
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(step 2) align the first side and add the T-shape on the four sides to form: (you can also see the video explanation in step 2)
In fact, you only need to learn one move to do this well. The blue character block has six positions after turning around,
For position a, just take the following three steps. For C, D, e and F, please go to the back to see how to turn them into a or B
For example, the blue, red and yellow corner block here must go up to the red and yellow corner, or the T word will not come out. Some friends may already think that I am like Tang Seng, but some friends do have this problem. I'd like to talk about it in detail
　　F D F'
In the initial state, this step is very ingenious, and achieves two purposes at the same time: one is to make the target corner position of the top layer to "connect" the blue, red and yellow corner blocks to the bottom layer, and the other is to make the blue, red and yellow corner blocks ready to the due position. Let the blue, red and yellow corner blocks turn to the target position and turn back to the top layer
And for position B, it's exactly the same, that is, make a mirror image of the top diagonal in the three steps just now
As shown in the figure on the left, for the top diagonal to make a mirror, we will encounter countless times later, so please pay attention to the significance of this mirror
The diagonal image of F D f 'above is r'd' R. please see,
This animation is not automatically played
Please press the play key to start r'd 'R
The graph of position B is the diagonal image of position a, so his algorithm is FDF 'looking at the diagonal mirror. The diagonal image of F is R'. You should understand it very well. See the picture of the mirror above. The image of D is the image of d 'f', which is R
For positions C, D, e and F, you can always turn them to position a or B by rotating the side and bottom surfaces
For F for C for D for e
If you can't watch the animation for the time being, you can click the algorithm below to see a simple illustration
　　F D2 F' D' F D F' D' R' D' R D F D F'
Repeat four corners and you'll get it
So we have laid the foundation, simple
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(step 3) put the second layer of edge color block into the shape (you can also see the video explanation in step 3)
From the previous step to this step, you can see that what we need to deal with in this step is the four edge color blocks of red white, red yellow, orange white and orange yellow in the middle layer. This time, you turn the magic cube over first, with the blue side down and the green side up, and then look at which of the four edge color blocks above are located on the top (the green centered side). Choose any one, and let's start from it. Here, for example, red and white, You have two situations,
We need to insert the top red and white edge color block into the position indicated by the red arrow in the figure above. You should put the corner between the red and white at the closest position to you (front right). Then one or two situations are very easy to identify. The row of the same color is in the front and the right respectively
For the first case, use the following algorithm,
　　U R U' R'
　　U' F' U F
The whole algorithm is uru'r '+ u'f'uf, isn't it very regular? Let's see, it seems that the second half of u'f'uf is just the diagonal image of the first half of uru'r', isn't it? Let's see in the first four steps, the function of uru'r 'is to combine the two small blocks we want in the left figure. The fifth figure marked red in the table above shows the two color blocks, The next four steps, u'f'uf, are to put our two pieces together in the right place
So, if you meet,
What should we do? Many friends may have seen that our algorithm should be the diagonal image of the above algorithm, because what we have to do is to look at the diagonal in a mirror, because the first half and the second half are diagonal images of each other, so the algorithm is to turn the first half and the second half upside down to become u'f'uf + uru'r ', the magic cube
　　U' F' U F
　　U R U' R'
In some cases, the edge you need is not on the top, but on the wrong position or orientation of the second layer. What should you do? First, you need to make the edge on the top, and the edge that may not be obedient will change to the top. If he is still not obedient, as shown in the figure on the left, we will use the above algorithm to get a irrelevant edge to that position, The red and white color block we want will naturally change to the top level, which is a little bit troublesome, but we have to bear with our elementary solution, which may also inspire us to learn advanced solutions
We have learned two algorithms so far, so we need to refuel
(step 4) draw a cross on the new top of the Rubik's cube (you can also see the video explanation in step 4)
Become:
Next, we need to learn a new algorithm. This algorithm will switch the top layer in the following four situations. After the four prism blocks on the top surface are rotated, there are only these four situations,
　　1 2 3 4
Probability 1 / 8 probability 1 / 4 probability 1 / 2 probability 1 / 8
In this step, let's turn the corner