Question

suppose that you have 4 green cards and 5 yellow cards. the cards are well shuffled. you randomly draw two cards with replacement. round your answers to four 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

Explanation:

Step1: Total cards calculation

Total cards = 4 + 5 = 9

Step2: P(G1) and P(G2) for a

P(G1) = 4/9, P(G2) = 4/9 (with replacement)

Step3: P(G1 and G2) calculation

P(G1 and G2) = (4/9)*(4/9) = 16/81 ≈ 0.1975

Step4: Complement for b

P(no green) = (5/9)*(5/9) = 25/81

Step5: P(at least 1 green) calculation

P(at least 1 green) = 1 - 25/81 = 56/81 ≈ 0.6914

Step6: Conditional probability for c

With replacement, P(G2|G1) = P(G2) = 4/9 ≈ 0.4444

Step7: Independence check for d

P(G2|G1) = P(G2), so independent

Answer:

a. 0.1975
b. 0.6914
c. 0.4444
d. They are independent events