Suppose that the probability of occurrence of event a in three repeated independent tests is 1 / 3, X is the number of occurrence of event a in three tests, and Y is the number of occurrence of event a in the first two tests Let the probability of occurrence of event a in three repeated independent tests be 1 / 3, X denotes the number of occurrence of event a in three tests, and Y denotes the number of occurrence of event a in the first two tests. It is difficult to find the joint distribution law of (x, y)

Suppose that the probability of occurrence of event a in three repeated independent tests is 1 / 3, X is the number of occurrence of event a in three tests, and Y is the number of occurrence of event a in the first two tests Let the probability of occurrence of event a in three repeated independent tests be 1 / 3, X denotes the number of occurrence of event a in three tests, and Y denotes the number of occurrence of event a in the first two tests. It is difficult to find the joint distribution law of (x, y)

All possible values of (x, y) are (0,0) (1,0) (1,1) (2,1) (2,2) (3,2)
P(0,0)=(2/3)³=8/27
P(1,0)=(2/3)²×(1/3)=4/27
P(1,1)=C2,1×1/3×(2/3)²=8/27
P(2,1)=C2,1×2/3×(1/3)²=4/27
P(2,2)=2/3×(1/3)²=2/27
P(3,2)=(1/3)³=1/27

Mathematical frequency and probability problems
The floor is covered with square tiles (40cm × 40cm). Now throw a circular dish with a radius of 5cm upward. What's the probability of the gap between the circular dish and the floor tile
Third volume (Beijing Normal University Edition) of a problem, please give the problem-solving process, thank you!

This is a typical topic of probability theory in University. The intersection of disc and brick seam is equivalent to that the distance between the center of disc and brick seam is less than the radius of circle. Then there is the problem of integration and measurement. If I use junior high school method, I will not. You can think about it according to the above equivalent conditions

There are three identical balls in the bag, which are marked with numbers, 1, 2, 3, respectively. Take a ball randomly from the bag, take the number on the ball as ten digits, put the ball back and shake it well, then take a ball randomly, and take the number on the ball as one digit. So, what's the probability that the number is 23?

The probability of extracting 2 in the first time is 1 / 3, and the probability of extracting 3 in the second time is 1 / 3, so the two must satisfy 1 / 9 of the multiplication at the same time

Xiao Ming did 80 mental arithmetic problems in 5 minutes, Xiao Dong did 120 mental arithmetic problems in 10 minutes. How many did Xiao Ming and Xiao Dong do in 1 minute

Here is the whole process:
80 △ 5 = 16 (question)
120 △ 10 = 12 (question)
16:12=4:3
A: the number of questions Xiao Ming and Xiao Dong do in one minute is 4:3

On the simple derivation of Lorentz transformation in a brief introduction to the theory of relativity
Two coordinate systems K and K ', if we take a snapshot at k' (t '= 0), and if we take a snapshot from the equation
x'=ax-bct
}
ct'=act-bx
If we take v = BC / a into account, we get
X '= a (1-V / C) C
I want to know how this result is derived
It's amazing. It's the wrong number
But I'd like to ask, why do you ask t instead of others?
Two results equal divisor and divisor at the same time square, the results are still equal, what is this theorem?

First of all, you have copied a wrong letter in the last equation, and the final factor is x, not C
Let t '= 0 in the second equation, and then t:
t=bx/ac
Substituting into the first equation, we get
x'=ax-bc*bx/ac=a(1-b²/a²)x=a(1-v²/c²)x
It's over

Xiao Ming did 12 mental arithmetic problems in three minutes, with an average of () per minute. Xiao Ming finished a few parts of these problems in one minute

Xiao Ming did 12 mental arithmetic problems in three minutes, with an average of (4) per minute. Xiao Ming finished one third of these problems in one minute

It is proved that the speed of light in any direction is constant C by Lorentz transformation

Now, based on these two facts, the transformation of coordinates is derived
Suppose that there are two inertial coordinate systems called s and s' respectively. The origin o 'of s' moves along the positive direction of X axis with respect to the origin o of s at the rate v. the space-time coordinates of any event in S and s' are (x, y, Z, t), (x', y ', Z', t '). When the two inertial systems coincide, the timing starts respectively. If x = 0, then x' + VT '= 0, Therefore, it is speculated that the transformation of the coordinates of any event from s' system to s system is x = γ (x '+ VT') (1), in which the constant γ is introduced and named as Lorentz factor (since this transformation is conjectured, it is obvious that it is necessary to carry out experiments to verify its correctness), That is to say, the physical equations in different inertial frames should have the same form. Therefore, the transformation of the above event coordinates from s-frame to s-frame to X '= γ (x-vt) (2) y and y', Z and Z 'can be directly obtained, that is, y' = y (3) Z '= Z (4) substituting (2) into (1), and solving t' to get t '= γ T + (1 - γ ^ 2) x / γ V (5), In order to find the value of γ, it is assumed that a beam of light in the positive direction along the x-axis is emitted from the coincident origin o (o '). Let the wavefront coordinates of the beam be (x, y, Z, t), (x', y ', Z', t '). According to the constant speed of light, x = CT (6) x' = CT '(7) (1) (2) multiplied by X' = γ ^ 2 (XX '- x'vt + xvt' - V ^ 2 * TT ') (8) take the event of wavefront as the object, Then (8) is written as XX '= γ ^ 2 (XX' - x'vt + xvt '- V ^ 2 * TT') (9) (6) (7) is substituted into (9), the Lorentz factor γ = [1 - (V / C) ^ 2] ^ (- 1 / 2) (10) (10) is substituted into (5), the T '= γ (t-vx / C ^ 2) (11) is put together, that is, the Lorentz transformation from s system to s' system X' = γ (x-vt), y '= y, Z '= Z, t' = γ (t-vx / C ^ 2) (12) according to the principle of relativity, the Lorentz transformation from s' system to s system x = γ (x '+ VT'), y = y ', z = Z', t = γ (t '+ VX' / C ^ 2) (13) is obtained. The velocity of a particle P is observed from s system and s' system respectively, n. M) and u '(J', n ', M'). The Lorentz velocity transformation from s' system to s' system is obtained by (12) deriving t '(J' = (J-V) / (1-vj / C ^ 2), n '= n / [γ (1-vj / C ^ 2) ^ - 1], M' = m / [γ (1-vj / C ^ 2) ^ - 1] (14) according to the relativity principle, the Lorentz velocity transformation from s' system to s' system is obtained by (14), j = (J '+ V) / (1 + VJ' / C ^ 2), n = n '/ [γ (1 + VJ' / C ^ 2) ^ - 1], M = m '/ [γ (1 + VJ' / C ^ 2) ^ - 1] (15) Lorentz transformation combined with momentum theorem and mass conservation law, all quantitative conclusions of special relativity can be obtained. After these conclusions are verified by experiments, it shows the correctness of special relativity

300 oral arithmetic problems in Grade 6
Be quick

2.8×0.4＝ 1.12
14－7.4＝6.6, 1.92÷0.04＝48, 0.32×500＝160, 0.65＋4.35＝ 5
10－5.4＝4.6, 4÷20＝0.2, 3.5×200＝700, 1.5－0.06＝1.44
0.75÷15＝0.05, 0.4×0.8＝0.32, 4×0.25＝1, 0.36＋1.54＝2
1.01×99＝99.99, 420÷35＝12, 25×12＝300, 135÷0.5＝270
2.8×0.4＝ 1.12
14－7.4＝6.6, 1.92÷0.04＝48, 0.32×500＝160, 0.65＋4.35＝ 5
10－5.4＝4.6, 4÷20＝0.2, 3.5×200＝700, 1.5－0.06＝1.44
0.75÷15＝0.05, 0.4×0.8＝0.32, 4×0.25＝1, 0.36＋1.54＝2
1.01×99＝99.99, 420÷35＝12, 25×12＝300, 135÷0.5＝270
3/4 + 1/4 =1, 2 + 4/9 =22/9, 3 - 2/3 =7/3, 3/4 - 1/2= 1/4
1/6 + 1/2 -1/6 =1/2, 7.5-(2.5+3.8)=1.2, 7/8 + 3/8 =5/4
3/10 +1/5 =1/2, 4/5 - 7/10 =1/10, 2 - 1/6 -1/3 =1.5
0.51÷17＝0.03, 32.8＋19＝51.8, 5.2÷1.3＝4, 1.6×0.4＝ 0.64
4.9×0.7＝3.43, 1÷5＝0.2, 6÷12＝0.5, 0.87－0.49＝0.38
2/1*2=1 3/1*3=1 3/2*3=2 3/1*6=2
4/3*8=6 5/3*20=12 7/3*14=6 8/7*40=35
4/3*16=12 9/5*27=15 2/1*30=15 12/7*24=14
30/1*30=1 51/9*102=18 19/9*76=36 4/9*8=18
5/8*90=144 99/98*99=98 3/14*6=28 7/1*28=4
10/1*90=9 5/3*105=63 19/7*38=14 5/1*25=5
8/19*16=38 61/60*122=120 7/2*28=8 6/1*48=8
9/7*18=14 25/7*100=28 9/5*81=45 8/9*16=18
2/1*2=1 3/1*3=1 3/2*3=2 3/1*6=2 4/3*8=6 5/3*20=12 7/3*14=6 8/7*40=35 4/3*16=12 9/5*27=15 2/1*30=15 12/7*24=14 30/1*30=1 51/9*102=18 19/9*76=36 4/9*8=18 5/8*90=144 99/98*99=98 3/14*6=28 7/1*28=4 10/1*90=9 5/3*105=63 19/7*38=14 5/1*25=5 8/19*16=38 61/60*122=120 7/2*28=8 6/1*48=8 9/7*18=14 25/7*100=28 9/5*81=45 8/9*16=18
12÷3/5=12×（ 5/3） 9÷6/7=9×（ 7/6 ） 30÷5/6=30×（6/5 ） 4×（3/2 ）=4÷2/3 （ 4 ）÷5/7=4×7/5 3÷4/5=3×5/4 （ 24 ）÷7/16=24×（16/7 ） A÷C/B=A×B/C 4÷4/5=5 6÷3/4=8 10÷2/5=25 18÷4/9=81/2 4×4/5=16/5 6×3/4=18/4 10×2/5=4 18×4/9=8 3÷3/4=4 2÷1/3=6 6÷4/5=15/2 1÷5/7=7/5 3/4÷3=1/4 1/3÷2=1/6 4/5÷6=2/15 5/7÷1=5/7
2/1*2=1 3/1*3=1 3/2*3=2 3/1*6=2
4/3*8=6 5/3*20=12 7/3*14=6 8/7*40=35
4/3*16=12 9/5*27=15 2/1*30=15 12/7*24=14
30/1*30=1 51/9*102=18 19/9*76=36 4/9*8=18
5/8*90=144 99/98*99=98 3/14*6=28 7/1*28=4
10/1*90=9 5/3*105=63 19/7*38=14 5/1*25=5
8/19*16=38 61/60*122=120 7/2*28=8 6/1*48=8
9/7*18=14 25/7*100=28 9/5*81=45 8/9*16=18
3/4 + 1/4 =1, 2 + 4/9 =22/9, 3 - 2/3 =7/3, 3/4 - 1/2= 1/4
1/6 + 1/2 -1/6 =1/2, 7.5-(2.5+3.8)=1.2, 7/8 + 3/8 =5/4
3/10 +1/5 =1/2, 4/5 - 7/10 =1/10, 2 - 1/6 -1/3 =1.5
0.51÷17＝0.03, 32.8＋19＝51.8, 5.2÷1.3＝4, 1.6×0.4＝ 0.64
4.9×0.7＝3.43, 1÷5＝0.2, 6÷12＝0.5, 0.87－0.49＝0.38
2/1*2=1 3/1*3=1 3/2*3=2 3/1*6=2
4/3*8=6 5/3*20=12 7/3*14=6 8/7*40=35
4/3*16=12 9/5*27=15 2/1*30=15 12/7*24=14
30/1*30=1 51/9*102=18 19/9*76=36 4/9*8=18
5/8*90=144 99/98*99=98 3/14*6=28 7/1*28=4
10/1*90=9 5/3*105=63 19/7*38=14 5/1*25=5
8/19*16=38 61/60*122=120 7/2*28=8 6/1*48=8
9/7*18=14 25/7*100=28 9/5*81=45 8/9*16=18
2/1*2=1 3/1*3=1 3/2*3=2 3/1*6=2 4/3*8=6 5/3*20=12 7/3*14=6 8/7*40=35 4/3*16=12 9/5*27=15 2/1*30=15 12/7*24=14 30/1*30=1 51/9*102=18 19/9*76=36 4/9*8=18 5/8*90=144 99/98*99=98 3/14*6=28 7/1*28=4 10/1*90=9 5/3*105=63 19/7*38=14 5/1*25=5 8/19*16=38 61/60*122=120 7/2*28=8 6/1*48=8 9/7*18=14 25/7*100=28 9/5*81=45 8/9*16=18
12÷3/5=12×（ 5/3） 9÷6/7=9×（ 7/6 ） 30÷5/6=30×（6/5 ） 4×（3/2 ）=4÷2/3 （ 4 ）÷5/7=4×7/5 3÷4/5=3×5/4 （ 24 ）÷7/16=24×（16/7 ） A÷C/B=A×B/C 4÷4/5=5 6÷3/4=8 10÷2/5=25 18÷4/9=81/2 4×4/5=16/5 6×3/4=18/4 10×2/5=4 18×4/9=8 3÷3/4=4 2÷1/3=6 6÷4/5=15/2 1÷5/7=7/5 3/4÷3=1/4 1/3÷2=1/6 4/5÷6=2/15 5/7÷1=5/7
2/1*2=1 3/1*3=1 3/2*3=2 3/1*6=2
4/3*8=6 5/3*20=12 7/3*14=6 8/7*40=35
4/3*16=12 9/5*27=15 2/1*30=15 12/7*24=14
30/1*30=1 51/9*102=18 19/9*76=36 4/9*8=18
5/8*90=144 99/98*99=98 3/14*6=28 7/1*28=4
10/1*90=9 5/3*105=63 19/7*38=14 5/1*25=5
8/19*16=38 61/60*122=120 7/2*28=8 6/1*48=8
From zhangtianxi 2008-12-20 13:06:33 0
2.8×0.4＝ 1.12
14－7.4＝6.6, 1.92÷0.04＝48, 0.32×500＝160, 0.65＋4.35＝ 5
10－5.4＝4.6, 4÷20＝0.2, 3.5×200＝700, 1.5－0.06＝1.44
0.75÷15＝0.05, 0.4×0.8＝0.32, 4×0.25＝1, 0.36＋1.54＝2
1.01×99＝99.99, 420÷35＝12, 25×12＝300, 135÷0.5＝270
2.8×0.4＝ 1.12
14－7.4＝6.6, 1.92÷0.04＝48, 0.32×500＝160, 0.65＋4.35＝ 5
10－5.4＝4.6, 4÷20＝0.2, 3.5×200＝700, 1.5－0.06＝1.44
0.75÷15＝0.05, 0.4×0.8＝0.32, 4×0.25＝1, 0.36＋1.54＝2
1.01×99＝99.99, 420÷35＝12, 25×12＝300, 135÷0.5＝270
3/4 + 1/4 =1, 2 + 4/9 =22/9, 3 - 2/3 =7/3, 3/4 - 1/2= 1/4
1/6 + 1/2 -1/6 =1/2, 7.5-(2.5+3.8)=1.2, 7/8 + 3/8 =5/4
3/10 +1/5 =1/2, 4/5 - 7/10 =1/10, 2 - 1/6 -1/3 =1.5
0.51÷17＝0.03, 32.8＋19＝51.8, 5.2÷1.3＝4, 1.6×0.4＝ 0.64
4.9×0.7＝3.43, 1÷5＝0.2, 6÷12＝0.5, 0.87－0.49＝0.38
2/1*2=1 3/1*3=1 3/2*3=2 3/1*6=2
4/3*8=6 5/3*20=12 7/3*14=6 8/7*40=35
4/3*16=12 9/5*27=15 2/1*30=15 12/7*24=14
30/1*30=1 51/9*102=18 19/9*76=36 4/9*8=18
5/8*90=144 99/98*99=98 3/14*6=28 7/1*28=4
10/1*90=9 5/3*105=63 19/7*38=14 5/1*25=5
8/19*16=38 61/60*122=120 7/2*28=8 6/1*48=8
9/7*18=14 2