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2 years ago

Which of the following transformations represents an isometry?

Mathematics

1 answer:

Wewaii [24]2 years ago

3 0

Answer:

Step-by-step explanation:

Isometry only includes, rotation, translation, and reflection.

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We have two coins, A and B. For each toss of coin A, we obtain Heads with probability 1/2 ; for each toss of coin B, we obtain H

ycow [4]

Answer:

The expected value is 5.

Step-by-step explanation:

Let X represent the number of tosses until the event described in the question happens.
Let Y represent the number of tosses with coin A until Heads is obtained.

Let Z represent the number of tosses with coin B until Heads is obtained.

As we can see, X=Y+Z. Then, by the linearity of the expected value operator, we have that

$E(X)=E(Y)+E(Z).$

We will compute E(Y) and E(Z).

Observe that Y and Z have countable sets of outcomes (1,2,3,....) then,

$E(X)=\sum^\infty_{n=1}nP(Y=n)$ ,

$E(Z)=\sum^\infty_{n=1}nP(Z=n)$ ,

Then:

for each $n\in \mathbb{N}$ , the probability of Y=n is given by $(0.5)^{n-1}(0.5)=(0.5)^{n}$ (because the first n-1 tosses must be Tails and the n-th must be Heads). Therefore

$E(Y)=\sum^\infty_{n=1}nP(Y=n)=\sum^\infty_{n=1}n(\frac{1}{2} )^n=\\\\\sum^\infty_{m=1}\sum^\infty_{n=m}(\frac{1}{2} )^n=\sum^\infty_{m=1}(\frac{1}{2} )^{m-1}=\sum^\infty_{m=0}(\frac{1}{2} )^{m}=2.$

For each $n\in \mathbb{N}$ , the probability of Z=n is given by $(\frac {2}{3})^{n-1}(\frac {1}{3})$ (because the first n-1 tosses must be Tails and the n-th must be Heads). Therefore

$E(Z)=\sum^\infty_{n=1}nP(Z=n)=\frac{1}{3}\sum^\infty_{n=1}n(\frac{2}{3} )^{n-1}=\frac{1}{3}\sum^\infty_{m=1}\sum^\infty_{n=m}(\frac{2}{3} )^{n-1}$

Observe that, by the geometric series formula:

$\sum^\infty_{n=m}(\frac{2}{3} )^{n-1}=\sum^\infty_{n=1}(\frac{2}{3} )^{n-1}-\sum^{m-1}_{n=1}(\frac{2}{3} )^{n-1}=3-\sum^{m-1}_{n=1}(\frac{2}{3} )^{n-1}=\\\\3-\sum^{m-2}_{n=0}(\frac{2}{3} )^{n}=3-\frac{1-(\frac{2}{3})^{m-1} }{1-\frac{2}{3}}=3(\frac{2}{3})^{m-1}$

Therefore

$E(Z)=\frac{1}{3}\sum^\infty_{m=1}\sum^\infty_{n=m}(\frac{2}{3} )^{n-1}=\frac{1}{3}\sum^\infty_{m=1}3(\frac{2}{3})^{m-1} =\\\\ \sum^\infty_{m=1}(\frac{2}{3})^{m-1} = \sum^\infty_{m=0}(\frac{2}{3})^{m} =3.$

Finally, E(X)=E(Y)+E(Z)=2+3=5.

4 0

2 years ago

Solve for y then find the value of x. y+6x=11;x=-1,0,3

Brrunno [24]

Answer:

Solution for y is y = 11 - 6x

For x = -1, the y value is 17

For x = 0, the y value is 11

For x = 3, the y value is -7

Step-by-step explanation:

Solve for y --> y + 6x = 11

y + 6x = 11

-6x -6x

------------------

y = 11 - 6x

Plug -1, 0, and 3 into this equation to find values of y --> x = 11 - y / 6

When x is -1, the y value becomes 17. Secondly, when x is 0, the y value becomes 11. Thirdly, when x is 3, the y value becomes -7.

3 0

2 years ago

Read 2 more answers

What is 1 1/3 divided by 3/5

enyata [817]

Answer:

55/9

Step-by-step explanation:

8 0

3 years ago

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Cuatro amigos van al cine dos veces al mes y siempre comen un paquete de palomitas de maiz cada uno si no comia palomitas de mai

EleoNora [17]

Answer:

El precio de la entrada al cine es de $56

Step-by-step explanation:

Cada uno de los 4 amigos las 2 veces por semana consumen un paquete de palomitas que cuesta: $28 ; entonces debemos multiplicar el costo de las palomitas por las dos veces que las consumen para saber el costo total de las palomitas:

28 x 2 =$56

El ejercicio nos dice que al no comer esas dos veces palomitas pueden ingresar a cine una tercera vez por semana ,lo que significa que al no pagar los $56 en palomitas podrán pagar una tercera entrada,lo que quiere decir que la entrada a cine es de $56.

Entonces podemos hacer la siguiente relación matemática: Precio total de las palomitas dos veces por semana = Precio de la entrada a cine.

7 0

2 years ago

4. What percent relates to when an event will definitely happen?

Pavel [41]

Answer:

Step-by-step explanation:

If we know that the event will definitely happen, then there is no doubt and no probability that the event will not happen.

This means that the probability is 1, which is the same as 100%.

6 0

3 years ago

Read 2 more answers