Question

Consider a bag that contains 231 coins of which 66

are rare Indian pennies. For the given pair of events A and​ B, complete parts​ (a) and​ (b) below.
​A: When one of the 231 coins is randomly​ selected, it is one of the 66 Indian pennies.
​B: When another one of the 231 coins is randomly selected​ (with replacement), it is also one of the
66 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.
Choose the correct answer below.

A. The two events are dependent because the​ 5% guideline indicates that they may be treated as dependent.
B.The two events are independent because the​ 5% guideline indicates that they may be treated as independent.
C.The two events are dependent because the occurrence of one affects the probability of the occurrence of the other.
D.The two events are independent because the occurrence of one does not affect the probability of the occurrence of the other.

Answer 1

Answer:

(a) Event A and B are independent.

(b) Probability = 0.0816

The correct answer from the choices:

D. The two events are independent because the occurrence of one does not affect the probability of the occurrence of the other

Step-by-step explanation:

Total coins = 231

Indian pennies in the the total coins = 66

Probability of Event A : Number of Indian pennies / total number of coins

66 / 231 = 0.2857

Probability of Event B : Number of Indian pennies / total number of coins

66 / 231 = 0.2857

(a) Since the coins are picked WITH replacement, the events are independent. As picking a coin does not change the probability of the next coin being picked. This is proven by how Event B has the same probability as Event A.

Thus, the answer to this part is D. 'The two events are independent because the occurrence of one does not affect the probability of the occurrence of the other.'

(b) The probability that both events occur will be their respective probabilities multiplied together. This is:

Total Probability = Probability of A * Probability of B

Total Probability = 0.2857 * 0.2857

Total Probability = 0.0816