Consider a bag that contains 220 coins of which 6 are rare Indian pennies. For the given pair of events A and​ B, complete parts​ (a) and​ (b) below. ​A: When one of the 220 coins is randomly​ selected, it is one of the 6 Indian pennies. ​B: When another one of the 220 coins is randomly selected​ (with replacement), it is also one of the 6 Indian pennies. a. Determine whether events A and B are independent or dependent. b. Find​ P(A and​ B), the probability that events A and B both occur.

Answer:a. The two events are dependent.b. [tex]P(A\cap B)[/tex]= [tex]\frac{1}{220}[/tex].Step-by-step explanation:Given Total coins =220Number of Indian pennies= 6A: When one of the 220 coins is randomly selected, it is one of the Indian pennies.Therefore , the probability of getting an  Indian pennies=[tex]\frac{6}{220 }[/tex]By using formula of probability=[tex]\frac{Number \; of\; favourable\; cases}{total\; number \; of \;cases}[/tex]Probability of getting an  Indian pennies=[tex]\frac{3}{110}[/tex]B: When another one of the 220 coins is randomly selected( with replacement) , It is also one of the Indian pennies.Therefore, probability of getting an Indian pennies=[tex]\frac{6}{220}[/tex]Probability of getting an Indian pennies =[tex]\frac{3}{110}[/tex][tex]A\cap B[/tex]: 1[tex]P(A\cap B)=\frac{1}{220}[/tex]If two events are independent. Then[tex]P(A\cap B)= P(A)\times p(B)[/tex]P(A).P(B)= [tex]\frac{3}{110} \times \frac{3}{110}[/tex]=[tex]\frac{9}{12100}[/tex]Hence, [tex]P(A\cap B)\neq P(A).P(B)[/tex]Therefore, the two events are dependent.b. Probability that events A and B both occurNumber of favourable cases when both events A and B occur=1Total coins=220Probability=[tex]\frac{Number \; of\; favourable \; cases}{Total\; number\; of\; cases}[/tex][tex]P(A\cap B)=\frac{1}{220}[/tex]