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Consider a bag that contains 225 coins of which 4 are rare Indian pennies. For the given pair of events A and​ B, complete parts​ (a) and​ (b) below. ​

A: When one of the 225 coins is randomly​ selected, it is one of the 4 Indian pennies. ​
B: When another one of the 225 coins is randomly selected​ (with replacement), it is also one of the 4 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.

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Answer:

a) Independent

b) P(A and​ B) = 0.000316 or 0.0316%

Step-by-step explanation:

a) Since it is specified that the selection of the coin in event B happens with replacement, the events are independent because each event does not interfere in the probability of the other. If there was no replacement, whether a rare coin was selected or not in event A would impact event B and then event B would depend on A.

b) For each event, there is a 4 in 225 chance of selecting a rare Indian penny. Therefore, P (A and B) is:

[tex]P(A\ and\ B) = P(A)*P(B)\\P(A\ and\ B) =\frac{4}{225}*\frac{4}{225} \\P(A\ and\ B) =0.000316 = 0.0316\%[/tex]

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