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2 years ago

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Bill placed a coordinate grid over a drawing of his back yard. Points A (-2, -3) and B (4, 9) represent two corners of his garde

n. He wants to place a water sprinkler (C) at the midpoint between A and B.

1 answer:

irinina [24]2 years ago

6 0

Answer:

(1;3)

Step-by-step explanation:

midpoint=(x1+x2;y1+y2)÷2

=(-2+4;-3+9)÷2

=(2;6)÷2

=(1;3)

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The prior probabilities for events A1 and A2 are P(A1) = 0.20 and P(A2) = 0.80. It is also known that P(A1 ∩ A2) = 0. Suppose P(

Umnica [9.8K]

Answer:

(a) $A_1$ and $A_2$ are indeed mutually-exclusive.

(b) $\displaystyle P(A_1\; \cap \; B) = \frac{1}{20}$ , whereas $\displaystyle P(A_2\; \cap \; B) = \frac{1}{25}$ .

(c) $\displaystyle P(B) = \frac{9}{100}$ .

(d) $\displaystyle P(A_1 \; |\; B) \approx \frac{5}{9}$ , whereas $P(A_1 \; |\; B) = \displaystyle \frac{4}{9}$

Step-by-step explanation:

<h3>(a)</h3>

$P(A_1 \; \cap \; A_2) = 0$ means that it is impossible for events $A_1$ and $A_2$ to happen at the same time. Therefore, event $A_1$ and $A_2$ are mutually-exclusive.

<h3>(b)</h3>

By the definition of conditional probability:

$\displaystyle P(B \; | \; A_1) = \frac{P(B \; \cap \; A_1)}{P(B)} = \frac{P(A_1 \; \cap \; B)}{P(B)}$ .

Rearrange to obtain:

$\displaystyle P(A_1 \; \cap \; B) = P(B \; |\; A_1) \cdot P(A_1) = 0.25 \times 0.20 = \frac{1}{20}$ .

Similarly:

$\displaystyle P(A_2 \; \cap \; B) = P(B \; |\; A_2) \cdot P(A_2) = 0.80 \times 0.05 = \frac{1}{25}$ .

<h3>(c)</h3>

Note that:

$\begin{aligned}P(A_1 \; \cup \; A_2) &= P(A_1) + P(A_2) - P(A_1 \; \cap \; A_2) = 0.20 + 0.80 = 1\end{aligned}$ .

In other words, $A_1$ and $A_2$ are collectively-exhaustive. Since $A_1$ and $A_2$ are collectively-exhaustive and mutually-exclusive at the same time:

$\displaystyle P(B) = P(B \; \cap \; A_1) + P(B \; \cap \; A_2) = \frac{1}{20} + \frac{1}{25} = \frac{9}{100}$ .

<h3>(d)</h3>

By Bayes' Theorem:

$\begin{aligned} P(A_1 \; |\; B) &= \frac{P(B \; | \; A_1) \cdot P(A_1)}{P(B)} \\ &= \frac{0.25 \times 0.20}{9/100} = \frac{0.05 \times 100}{9} = \frac{5}{9}\end{aligned}$ .

Similarly:

$\begin{aligned} P(A_2 \; |\; B) &= \frac{P(B \; | \; A_2) \cdot P(A_2)}{P(B)} \\ &= \frac{0.05 \times 0.80}{9/100} = \frac{0.04 \times 100}{9} = \frac{4}{9}\end{aligned}$ .

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2 years ago

1/2r -3= 3 (4-3/2r) solve for r

s2008m [1.1K]

1/2r -3= 3 (4-3/2r) is to be solved for r.

I'll begin by making the assumptions that by 1/2r you actually meant (1/2)r and that by 3/2r you actually meant (3/2)r. When in doubt, please use parentheses to make your meaning clear.

Thus, 1/2r -3= 3 (4-3/2r) becomes (1/2)r -3= 3 (4-(3/2)r ) .

Simplify this by multiplying all 3 terms by 2. Doing this will eliminate the fractions:

r -6 = 3 (4*2-(3)r ) or r - 6 = 24 - 9r

Now expand the right side, using the distributive property of

r - 3 = 24 - 9r

Regrouping so as to combine like terms:

10r = 30

Solving for r: r = 30/10 = 3

The value of r that satisfies this equation is 3

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