Question

c) after the first marble is drawn, for the second draw, how many marbles are left in the jar? what is the probability of drawing:
d) a red marble on the second draw, given that a red marble was drawn first? hint: be careful with the denominator. how many marbles are in the jar?
e) a blue marble on the second draw, given that a red marble was drawn first?
for the remaining parts of this problem, instead suppose that the first marble is replaced (returned to the jar) before the second marble is drawn.
f) after the first marble is drawn, for the second draw, how many marbles are left in the jar? what is the probability of drawing:
g) a red marble on the second draw, given that a red marble was drawn first?
h) a blue marble on the second draw, given that a red marble was drawn first?

Explanation:

Step1: Understand the non - replacement case

When a red marble is drawn first in non - replacement, one red marble is removed. Let the initial number of red marbles be $r$ and total marbles be $n$. After drawing one red marble, the number of red marbles left is $r - 1$ and total marbles left is $n-1$.

Step2: Calculate probability for non - replacement (e)

If a red marble is drawn first, and we want to draw a blue marble second. Let the initial number of blue marbles be $b$. After drawing one red marble (non - replacement), total marbles are $n - 1$. So the probability $P=\frac{b}{n - 1}$. If initially there are 31 blue marbles and after first draw (red) there are 55 marbles left, the correct probability of drawing a blue marble second given red first (non - replacement) is $\frac{31}{55}$.

Step3: Understand the replacement case

When the first marble is replaced, the total number of marbles and the number of each color of marbles remain the same for the second draw as they were for the first draw.

Step4: Calculate probability for replacement (g)

If initially there are $r$ red marbles and $n$ total marbles, and the first red marble is replaced, the probability of drawing a red marble second given a red marble first is $\frac{r}{n}$.

Step5: Calculate probability for replacement (h)

If initially there are $b$ blue marbles and $n$ total marbles, and the first red marble is replaced, the probability of drawing a blue marble second given a red marble first is $\frac{b}{n}$.

Answer:

e) $\frac{31}{55}$
g) Let the initial number of red marbles be $r$ and total marbles be $n$, probability is $\frac{r}{n}$
h) Let the initial number of blue marbles be $b$ and total marbles be $n$, probability is $\frac{b}{n}$