suppose that you have 7 green cards and 5 yellow cards. the cards are well shuffled. you randomly draw two…

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.\ng1 = the first card drawn is green\ng2 = the second card drawn is green\na. p(g1 and g2) =\n\nb. p(at least 1 green) =\n\nc. p(g2|g1) =\n\nd. are g1 and g2 independent?\n\nthey are independent events\nthey are dependent events\nhint: independent events\nvideo on independent events ·

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.\ng1 = the first card drawn is green\ng2 = the second card drawn is green\na. p(g1 and g2) =\n\nb. p(at least 1 green) =\n\nc. p(g2|g1) =\n\nd. are g1 and g2 independent?\n\nthey are independent events\nthey are dependent events\nhint: independent events\nvideo on independent events ·

Answer

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