Question

suppose that you have 7 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
hint: independent events
video on independent events ·

Explanation:

Step1: Calculate total number of cards

Total cards = 7 (green) + 5 (yellow)=12.

Step2: Calculate P(G1 and G2)

Since draws are with - replacement, P(G1) = P(G2)=$\frac{7}{12}$. By the multiplication rule for independent events, P(G1 and G2)=P(G1)×P(G2)=$\frac{7}{12}\times\frac{7}{12}=\frac{49}{144}\approx0.3403$.

Step3: Calculate P(At least 1 green)

First, find P(no green). P(no green) = P(both yellow). P(yellow)=$\frac{5}{12}$, so P(both yellow)=$\frac{5}{12}\times\frac{5}{12}=\frac{25}{144}$. Then P(At least 1 green)=1 - P(no green)=1 - $\frac{25}{144}=\frac{119}{144}\approx0.8264$.

Step4: Calculate P(G2|G1)

Since the draws are with - replacement, the outcome of the first draw does not affect the second draw. So P(G2|G1)=P(G2)=$\frac{7}{12}\approx0.5833$.

Step5: Determine independence

Since P(G2|G1)=P(G2) and P(G1 and G2)=P(G1)×P(G2), G1 and G2 are independent events.

Answer:

a. 0.3403
b. 0.8264
c. 0.5833
d. They are independent events