Please Help! A bag contains four green marbles and five yellow marbles. You randomly select three marbles. What is the probability that all three marbles are green when (a) you replace each marble before selecting the next marble, and (b) you do not replace each marble before selecting the next marble? Write each probability as a decimal rounded to the nearest thousandth. Then compare the probabilities.

A. 0.088

B. 0.048

C. 1.83 times

Step-by-step explanation:

From the question given above, the following data were obtained:

Green (G) = 4

Yellow (Y) = 5

Total = 4 + 5 = 9

A. Determination of the probability of selecting three green marbles with replacement.

Green (G) = 4

Yellow (Y) = 5

Total = 9

P(1st G) = 4/9

P(2nd G) = 4/9

P(3rd G) = 4/9

P(all G) = P(1st G) × P(2nd G) × P(3rd G)

P(all G) = 4/9 × 4/9 × 4/9

P(all G) = 64/729

P(all G) = 0.088

B. Determination of the probability of selecting three green marbles without replacement.

Green (G) = 4

Yellow (Y) = 5

Total = 9

P(1st G) = 4/9

P(2nd G) = 3/8

P(3rd G) = 2/7

P(all G) = P(1st G) × P(2nd G) × P(3rd G)

P(all G) = 4/9 × 3/8 × 2/7

P(all G) = 24/504

P(all G) = 0.048

C. Comparing probability of replacement and without replacement.

P(all G with replacement) = A = 0.088

P(all G without replacement) = B = 0.048

A/B = 0.088/0.048

A/B = 1.83

Cross multiply

A = 1.83 × B

Thus,

P(all G with replacement) = 1.83 × P(all G without replacement)

  5.   y=6; de=7; fg=8

  6.   x=4.2; jl=14.2; mn=13

step-by-step explanation:

when chords cross inside a circle, the product of the segment lengths is a constant.

5. dh×eh = fh×gh

  y = gh = dh×eh/fh = 3×4/2 . . . divide by the coefficient of gh

  y = 6

of course, the total chord length is the sum of the lengths of its segments.

  de = dh +eh = 3+4 = 7

  fg = fh +gh = 2+6 = 8

6. this problem works the same way as the previous one.

  x = lk = mk×nk/jk = 6×7/10 = 4.2

  jl = jk +lk = 10 +4.2 = 14.2

  mn = mk +nk = 6 +7 = 13

comment on chords and secants

you can think of the points h (problem 5) and k (problem 6) as points where the chords meet. a chord is part of a secant line, a line that intersects the circle in two points. one can also consider these points (h or k) to be the points where the respective secant lines meet.

then, the product that is a constant is the product of the distance from the meeting point (h or k) to one circle intersection with the distance from that meeting point to the other circle intersection. for problem 5, the constant is the product hd×he = hf×hg.

it turns out that this rule regarding the products of lengths to points of intersection is also true when the secants intersect outside the circle.

that is, you only need to remember one rule to work both kinds of problems.

step-by-step explanation:

12: 00 p.m. to 7: 00 p.m. is a total of 7 hours.

4: 00 p.m. to 7: 00 p.m. is a total of 3 hours.

this means that from 4: 00 p.m to 7: 00 p.m. this represents 3/7 of what was lost from 12: 00 p.m. to 7: 00 p.m.

after knowing this, after doing 11,600 - 6,800 = 4,800

4,800 represents 3/7 of what was lost for 7 hours.

we then divide 4,800 by 3 which gets 1,600.

because we know that 1,600 represents 1/7 of how much water was lost, we just have to multiply by 7, which gets 11,200.   after adding 11,200 to how much water was left at 7: 00 p.m. which was 6,800. 11,200 + 6,800 = 18,000

this means that the pool originally had 18,000 gallons in the pool.

you can also check your work by doing 18,000 - 4(1600), which becomes           1800 - 6400 which gets you 11,600.