Question

marbles problems

	glass	steel	plastic	total
red	6	13	20	39
blue	1	11	12	24
total:	22	27	32	81

2 fill in the blank 3 points
find the probability of drawing a glass marble. p(glass) = choose your answer...
find the probability that a random marble from the above jar is glass, given that its green. p(glass | green) = choose your answer...
are events \glass\ and \green\ independent events? choose your answer...

Explanation:

Step1: Calculate P(glass)

The probability of an event is the number of favorable outcomes divided by the total number of outcomes. The number of glass marbles is 22 and the total number of marbles is 81. So $P(glass)=\frac{22}{81}$.

Step2: Calculate P(glass|green)

The formula for conditional - probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Here, $A$ is the event of drawing a glass marble and $B$ is the event of drawing a green marble. $P(A\cap B)$ is the probability of drawing a green glass marble, which is $\frac{15}{81}$, and $P(B)$ is the probability of drawing a green marble, which is $\frac{18}{81}$. So $P(glass|green)=\frac{P(glass\cap green)}{P(green)}=\frac{\frac{15}{81}}{\frac{18}{81}}=\frac{15}{18}=\frac{5}{6}$.

Step3: Check for independence

Two events $A$ and $B$ are independent if $P(A|B) = P(A)$. We have $P(glass)=\frac{22}{81}\approx0.272$ and $P(glass|green)=\frac{5}{6}\approx0.833$. Since $P(glass)
eq P(glass|green)$, the events "glass" and "green" are not independent.

Answer:

$P(glass)=\frac{22}{81}$
$P(glass|green)=\frac{5}{6}$
No, the events "glass" and "green" are not independent events.