Consider a bag that contains 218 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 218 coins is randomly​ selected, it is one of the 6 Indian pennies. ​B: When another one of the 218 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.

a. To determine whether events A and B are independent or dependent, we need to understand what each term means. Two events are considered independent if the occurrence of one event does not affect the probability of the occurrence of the other event. On the other hand, events are dependent if the occurrence of one event affects the probability of the other event happening.

In this case, the problem states that after selecting the first coin (event A), the coin is replaced before selecting the second coin (event B). This means the first selection does not affect the second selection because the total number of coins and the number of Indian pennies available for the second selection remains the same. Therefore, events A and B are independent.

b. Since events A and B are independent, we can find the probability of both A and B occurring (P(A and B)) by multiplying the probabilities of each event.

First, we calculate the probability of event A (selecting an Indian penny):

P(A) = (Number of Indian pennies) / (Total number of coins) = 6/218

Next, since the coin is replaced, the probability of event B (again selecting an Indian penny) is the same as that for event A:

P(B) = (Number of Indian pennies) / (Total number of coins) = 6/218

Now we multiply these probabilities together to find P(A and B):

P(A and B) = P(A) * P(B) = (6/218) * (6/218)

When we multiply these two fractions, we get:

P(A and B) = (6 * 6) / (218 * 218) = 36 / 47524

Using a calculator, we can simplify this fraction if necessary, but it gives us the exact probability of both A and B occurring.

Extra: Understanding Probability - Probability is the measure of the likelihood of an event to occur. It is a number between 0 and 1, where 0 indicates impossibility, and 1 indicates certainty. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Independent Events - As we saw in the example, independent events do not influence each other's outcomes. Another example of independent events would be flipping a coin and rolling a die. The outcome of the coin flip doesn't affect the result of the die roll and vice versa.

Dependent Events - An example of dependent events would be drawing cards from a deck without replacement. If you draw one card, the deck now has one less card, which affects the probability of drawing a second card of a particular kind.

Calculating Combined Probabilities - When calculating the probability of two independent events both occurring, you multiply the probability of the first event by the probability of the second event. This rule only applies to independent events. For dependent events, the calculation would consider how the outcome of the first event affects the second event.