Mathematics, 21.06.2021 15:50 landowatson123

A box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Find the probability of the following events: a. Let F = the event of getting the white ball twice.
b. Let G = the event of getting two balls of different colors.
c. Let H = the event of getting white on the first pick.
d. Are F and G mutually exclusive?
e. Are G and H mutually exclusive?

Answers: 3

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Answers

Answer from: trodmanmalachip2yn26

See explanation

Step-by-step explanation:

Given

Represent the balls with the first letters

W =1

R =1

Solving (a): P(F) --- White balls twice

The event of F is:

$F = \{(W,W)\}$

So:

P(F) = P(W) * P(W)

$P(F) = \frac{n(W)}{n} * \frac{n(W)}{n}$

$P(F) = \frac{1}{2} * \frac{1}{2}$

$P(F) = \frac{1}{4}$

Solving (b): P(G) --- two different colors

The event of G is:

$G = \{(W,R),(R,W)\}$

So:

P(G) = P(W) * P(R) + P(R) * P(W)

$P(G) = \frac{n(W)}{n} * \frac{n(R)}{n} + \frac{n(R)}{n} * \frac{n(W)}{n}$

$P(G) = \frac{1}{2} * \frac{1}{2} + \frac{1}{2} * \frac{1}{2}$

$P(G) = \frac{1}{4} + \frac{1}{4}$

$P(G) = \frac{1}{2}$

Solving (c): P(H) --- White picked first

The event of H is:

$H = \{(W,R),(W,W)\}$

So:

P(H) = P(W) * P(R) + P(W) * P(W)

$P(H) = \frac{n(W)}{n} * \frac{n(R)}{n} + \frac{n(W)}{n} * \frac{n(W)}{n}$

$P(H) = \frac{1}{2} * \frac{1}{2} + \frac{1}{2} * \frac{1}{2}$

$P(H) = \frac{1}{4} + \frac{1}{4}$

$P(H) = \frac{1}{2}$

Solving (d): F and G, mutually exclusive?

We have:

$F = \{(W,W)\}$

$G = \{(W,R),(R,W)\}$

Check for common elements

$n(F\ n\ G) = 0$

Hence, F and G are mutually exclusive

Solving (e): G and G, mutually exclusive?

We have:

$G = \{(W,R),(R,W)\}$

$H = \{(W,R),(W,W)\}$

Check for common elements

$n(G\ n\ H) = 1$

Hence, F and G are not mutually exclusive

Answer from: Quest

part a

answer: a reasonable domain is $0 \le n \le 10$ where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

=====================================

part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

=====================================

part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).

Answer from: Quest

answer: sensy

step-by-step explanation: