Final answer:
The probability of Event C occurring is independent since the outcome of rolling a die and the outcome of the spinner do not affect each other. Calculating probabilities for other events like A and B involves counting outcomes and determining exclusivity or independence based on common outcomes and the product of probabilities.
Step-by-step explanation:
The probability of Event C occurring, which is rolling a number less than 6 and the spinner landing on yellow, is dependent on the outcomes of two separate random devices: a die and a spinner. To determine if these events are independent, we must assess whether the outcome of one event affects the other. However, from the information given, it appears the events are independent because the outcome of the die roll does not seem to influence the outcome of the spinner. Without further context, the statement that Event C is independent could be considered true.
- Event A involves rolling a three or a four on the die, followed by rolling an even number. To find P(A), we consider the probability of rolling a three or a four (2 out of 6 possibilities) and the probability of rolling an even number on the second roll (3 out of 6 possibilities).
- P(B) involves the sum of two die rolls being at most seven. We count the number of outcomes where the sum is 7 or less and divide by the total number of possible outcomes (36).
To determine if events A and B are mutually exclusive or independent, we analyze if they can occur simultaneously without influencing each other's outcomes. For mutual exclusivity, we look for any common outcomes. For independence, we check if P(A and B) = P(A)P(B).
The concept of mutually exclusive events is illustrated in the scenario of a coin toss where Event C (all heads) and Event B (all tails) cannot occur simultaneously, making them mutually exclusive. Independence between two events is showcased when the outcome of one event does not affect the outcome of another, exemplified by rolling a specific number on a die and then tossing a coin, where the two actions do not influence each other.