You have a large jar that initially contains 30 red marbles and 20 blue marbles. we also have a large supply of extra marbles of each color. draw a marble out of the jar. if it's red, put it back in the jar, and add three red marbles to the jar from the supply of extras. if it's blue, put it back into the jar, and add five blue marbles to the jar from the supply of extras. do this two more times. now, pull a marble from the jar, at random. what's the probability that this last marble is red? what's the probability that we actually drew the same marble all four times?

There is a 57.68% probability that this last marble is red.

There is a 20.78% probability that we actually drew the same marble all four times.

Step-by-step explanation:

Initially, there are 50 marbles, of which:

30 are red

20 are blue

Any time a red marble is drawn:

The marble is placed back, and another three red marbles are added

Any time a blue marble is drawn

The marble is placed back, and another five blue marbles are added.

The first three marbles can have the following combinations:

R - R - R

R - R - B

R - B - R

R - B - B

B - R - R

B - R - B

B - B - R

B - B - B

Now, for each case, we have to find the probability that the last marble is red. So

is the probability that we go R - R - R - R

There are 50 marbles, of which 30 are red. So, the probability of the first marble sorted being red is .

Now the red marble is returned to the bag, and another 3 red marbles are added.

Now there are 53 marbles, of which 33 are red. So, when the first marble sorted is red, the probability that the second is also red is

Again, the red marble is returned to the bag, and another 3 red marbles are added

Now there are 56 marbles, of which 36 are red. So, in this sequence, the probability of the third marble sorted being red is

Again, the red marble sorted is returned, and another 3 are added.

Now there are 59 marbles, of which 39 are red. So, in this sequence, the probability of the fourth marble sorted being red is . So

is the probability that we go R - R - B - R

is the probability that we go R - B - R - R

is the probability that we go R - B - B - R

is the probability that we go B - R - R - R

is the probability that we go B - R - B - R

is the probability that we go B - B - R - R

is the probability that we go B - B - B - R

So, the probability that this last marble is red is:

There is a 57.68% probability that this last marble is red.

What's the probability that we actually drew the same marble all four times?

is the probability that we go R-R-R-R. It is the same from the previous item(the last marble being red). So

is the probability that we go B-B-B-B. It is almost the same as in the previous exercise. The lone difference is that for the last marble we want it to be blue. There are 65 marbles, 35 of which are blue.

There is a 20.78% probability that we actually drew the same marble all four times

answer: 2x + 2 − 2 = 10 − 2; subtraction property of equality

step-by-step explanation:

answer: c, the green line

step-by-step explanation:

there is no black line, eliminate b.

simple interest implies periodic payments, but the red line shows no payments, eliminate d.

simple interest has equal payments. if paid say every day, the graph is about a straight line (actually a stairstep).

the yellow line is not straight, eliminate a.

check: the green line is straight, answer is c.

simple interest with positive interest rate gives a straight line sloping up to the right. green line:

g(t) = $100 + ($200-$100) ÷ (20 years) × (t years)

= $100 + (4%/year) × ($100)(t years)

incidentally, the yellow line curving upward represents continuously compounded interest

y = $100 × 2^(t years / (14 assuming the yellow line hits $200 at 14 years. 2^0 is 1, so y(0) = $100; 2^(14/14) is 2, so y(14) = $200.