Question

suppose that you have 4 green cards and 5 yellow cards. the cards are well shuffled. you randomly draw two cards without replacement. round answers to 4 decimal places.
g1 = the first card drawn is green
g2 = the second card drawn is green
a. p(g1 and g2) =

b. p(at least 1 green) =

c. p(g2|g1) =

d. are g1 and g2 independent?

they are independent events
they are dependent events
hint: independent events
video on independent events +

Explanation:

Step1: Calculate total number of cards

Total cards = 4 (green) + 5 (yellow)=9 cards.

Step2: Calculate P(G1 and G2)

P(G1)=$\frac{4}{9}$, after drawing a green - card first, there are 3 green cards left and 8 total cards left. So P(G2|G1)=$\frac{3}{8}$. By the multiplication rule P(G1 and G2)=P(G1)×P(G2|G1)=$\frac{4}{9}\times\frac{3}{8}=\frac{12}{72}\approx0.1667$.

Step3: Calculate P(At least 1 green)

P(At least 1 green)=1 - P(no green). P(no green) means drawing two yellow cards. P(first yellow)=$\frac{5}{9}$, P(second yellow|first yellow)=$\frac{4}{8}$. So P(no green)=$\frac{5}{9}\times\frac{4}{8}=\frac{20}{72}$. Then P(At least 1 green)=1 - $\frac{20}{72}=\frac{52}{72}\approx0.7222$.

Step4: Calculate P(G2|G1)

As calculated in Step 2, if the first card is green, there are 3 green cards and 8 total cards left. So P(G2|G1)=$\frac{3}{8}=0.3750$.

Step5: Determine independence

Two events G1 and G2 are independent if P(G2|G1)=P(G2). P(G2)=$\frac{4}{9}$ (before any card is drawn), and P(G2|G1)=$\frac{3}{8}$. Since P(G2|G1)≠P(G2), G1 and G2 are dependent events.

Answer:

a. 0.1667
b. 0.7222
c. 0.3750
d. They are dependent events