A girl with a taste for intricate schemes has 3 lavender and 2 red marbles in...

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A girl with a taste for intricate schemes has 3 lavender and 2 red marbles in her left pocket, and 1 lavender and 4 red marbles in her right pocket. She chooses one of her pockets (called the "chosen pocket") by drawing a randomly chosen marble D from her left pocket: if D is Lavender the Left pocket is the chosen pocket, and if D is Red the Right pocket is the chosen pocket. She returns D to her left pocket. She then draws a randomly chosen marble Mị from the chosen pocket, notes its color, returns it to the same pocket, and then again draws a randomly chosen marble M2 from the same pocket. Let L, be the event that M is lavender and L2 be the event that M2 is lavender. Give exact answers as simplified fractions. (a) Calculate P(L) and P(L2). (b) Calculate P(L2 | L1). (Hint: you cannot assume without proof that L and La are independent.) (c) Are L1 and L2 are independent? Justify your answer. (d) If both M1 and M2 are lavender, what is the probability that D was lavender? A girl with a taste for intricate schemes has 3 lavender and 2 red marbles in her left pocket, and 1 lavender and 4 red marbles in her right pocket. She chooses one of her pockets (called the "chosen pocket") by drawing a randomly chosen marble D from her left pocket: if D is Lavender the Left pocket is the chosen pocket, and if D is Red the Right pocket is the chosen pocket. She returns D to her left pocket. She then draws a randomly chosen marble Mị from the chosen pocket, notes its color, returns it to the same pocket, and then again draws a randomly chosen marble M2 from the same pocket. Let L, be the event that M is lavender and L2 be the event that M2 is lavender. Give exact answers as simplified fractions. (a) Calculate P(L) and P(L2). (b) Calculate P(L2 | L1). (Hint: you cannot assume without proof that L and La are independent.) (c) Are L1 and L2 are independent? Justify your answer. (d) If both M1 and M2 are lavender, what is the probability that D was lavender?