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andrey2020 [161]
3 years ago
13

Find the probability of rolling a prime number on the first die and even number on the second

Mathematics
1 answer:
Zinaida [17]3 years ago
4 0

Answer:

0.25

Step-by-step explanation:

Prime outcomes on a die: 2,3,5

P(prime) = 3/6 = 1/2

Even outcom: 2,4,6

P(even) = 3/6 = 1/2

P(prime & even)

= P(prime) × P(even)

= ½ × ½

= ¼ or 0.25

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Binomial Expansion/Pascal's triangle. Please help with all of number 5.
Mandarinka [93]
\begin{matrix}1\\1&1\\1&2&1\\1&3&3&1\\1&4&6&4&1\end{bmatrix}

The rows add up to 1,2,4,8,16, respectively. (Notice they're all powers of 2)

The sum of the numbers in row n is 2^{n-1}.

The last problem can be solved with the binomial theorem, but I'll assume you don't take that for granted. You can prove this claim by induction. When n=1,

(1+x)^1=1+x=\dbinom10+\dbinom11x

so the base case holds. Assume the claim holds for n=k, so that

(1+x)^k=\dbinom k0+\dbinom k1x+\cdots+\dbinom k{k-1}x^{k-1}+\dbinom kkx^k

Use this to show that it holds for n=k+1.

(1+x)^{k+1}=(1+x)(1+x)^k
(1+x)^{k+1}=(1+x)\left(\dbinom k0+\dbinom k1x+\cdots+\dbinom k{k-1}x^{k-1}+\dbinom kkx^k\right)
(1+x)^{k+1}=1+\left(\dbinom k0+\dbinom k1\right)x+\left(\dbinom k1+\dbinom k2\right)x^2+\cdots+\left(\dbinom k{k-2}+\dbinom k{k-1}\right)x^{k-1}+\left(\dbinom k{k-1}+\dbinom kk\right)x^k+x^{k+1}

Notice that

\dbinom k\ell+\dbinom k{\ell+1}=\dfrac{k!}{\ell!(k-\ell)!}+\dfrac{k!}{(\ell+1)!(k-\ell-1)!}
\dbinom k\ell+\dbinom k{\ell+1}=\dfrac{k!(\ell+1)}{(\ell+1)!(k-\ell)!}+\dfrac{k!(k-\ell)}{(\ell+1)!(k-\ell)!}
\dbinom k\ell+\dbinom k{\ell+1}=\dfrac{k!(\ell+1)+k!(k-\ell)}{(\ell+1)!(k-\ell)!}
\dbinom k\ell+\dbinom k{\ell+1}=\dfrac{k!(k+1)}{(\ell+1)!(k-\ell)!}
\dbinom k\ell+\dbinom k{\ell+1}=\dfrac{(k+1)!}{(\ell+1)!((k+1)-(\ell+1))!}
\dbinom k\ell+\dbinom k{\ell+1}=\dbinom{k+1}{\ell+1}

So you can write the expansion for n=k+1 as

(1+x)^{k+1}=1+\dbinom{k+1}1x+\dbinom{k+1}2x^2+\cdots+\dbinom{k+1}{k-1}x^{k-1}+\dbinom{k+1}kx^k+x^{k+1}

and since \dbinom{k+1}0=\dbinom{k+1}{k+1}=1, you have

(1+x)^{k+1}=\dbinom{k+1}0+\dbinom{k+1}1x+\cdots+\dbinom{k+1}kx^k+\dbinom{k+1}{k+1}x^{k+1}

and so the claim holds for n=k+1, thus proving the claim overall that

(1+x)^n=\dbinom n0+\dbinom n1x+\cdots+\dbinom n{n-1}x^{n-1}+\dbinom nnx^n

Setting x=1 gives

(1+1)^n=\dbinom n0+\dbinom n1+\cdots+\dbinom n{n-1}+\dbinom nn=2^n

which agrees with the result obtained for part (c).
4 0
3 years ago
Each day, Karen makes 37 loaves of bread at the bakery where she works. Each loaf of bread requires 6 cups of flour. One pound o
Vika [28.1K]
55.5
37 loaves at 6 cups each is 222 cups
4 cups to a pound means dividing the 222 cups by 4 leaves you with 55.5 pounds
5 0
3 years ago
Read 2 more answers
Write an equation of the line containing the point (2,1) and perpendicular to the line 5x – 2y = 3.
Ann [662]

So first, you want to isolate your Y. To do this, you must get it alone on ONE SIDE of the equation.

5x - 2y = 3

-5x         -5x

\frac{-2y}{-2} = \frac{3-5x}{-2}

 

ANSWER: y = \frac{3-5x}{-2}

6 0
3 years ago
Find the distance between the points A(2,-1) and B(-4, 2). (simplify your answer)
Pani-rosa [81]

Answer:

d = 3\sqrt{5}

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right

<u>Algebra II</u>

  • Distance Formula: d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Step-by-step explanation:

<u>Step 1: Define</u>

Point A (2, -1)

Point B (-4, 2)

<u>Step 2: Find distance </u><em><u>d</u></em>

Simply plug in the 2 coordinates into the distance formula to find distance <em>d</em>.

  1. Substitute [DF]:                    d = \sqrt{(-4-2)^2+(2+1)^2}
  2. Subtract/Add:                       d = \sqrt{(-6)^2+(3)^2}
  3. Exponents:                           d = \sqrt{36+9}
  4. Add:                                      d = \sqrt{45}
  5. Simplify:                               d = 3\sqrt{5}
3 0
3 years ago
Convert 32.5oz to its equivalent in cg
Black_prince [1.1K]
32,5
1oz=1000cg 
32.5*1000=32500cg

3 0
3 years ago
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