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In probability theory, independent events are those where the occurrence of one event does not influence the occurrence of the other. This concept is crucial because it can simplify probability calculations. To check whether two events are independent, we need to look at their individual probabilities and see if they relate to the combined probability of both events occurring together.

For example, if Event A has a probability of occurring as \(P(A)\), and Event B has a probability \(P(B)\), the two events are independent if the probability of both occurring is the product of their individual probabilities: \(P(A \cap B) = P(A) \times P(B)\).

In the case of Spike, we see that the probability that he passes both chemistry and mathematics together is given as \(0.12\). When checking their independence, the calculation of individual probabilities product \( (0.35 \times 0.40 = 0.14) \) shows a discrepancy with this value. Since \(0.12 eq 0.14\), we conclude that these events are not independent. Therefore, the outcome of passing one subject does indeed affect the probability of passing the other.

Calculating probabilities often involves a clear understanding of individual events and how they can occur together or separately.

In solving problems related to probability, one common approach involves using known values and relations. In Spike's exercise, we used basic probability principles to determine independence and to calculate other probabilities, such as the likelihood of failing both subjects.

• Passing chemistry: \(0.35\)
• Passing mathematics: \(0.40\)
• Passing both subjects: \(0.12\)

To determine if events are independent, we reviewed whether the probability of both events happening together equals the product of their individual probabilities. This comparison provides insight into whether the events influence each other. Since independence was not met, further calculations took this relationship into account.

Failure probability is the chance that a desired outcome will not occur. For any event A, the failure probability can be derived as the complement of its success probability and is given by \(1 - P(A)\).

Calculating the failure probability of multiple events such as Spike failing both subjects requires an understanding of their dependence. Since the events "Spike passes chemistry" and "Spike passes mathematics" are not independent, this must be accounted for when calculating the joint failure probability.

To calculate the probability that Spike fails both subjects:

• Find the probability of failing chemistry: \(1 - 0.35 = 0.65\)
• Find the probability of failing mathematics: \(1 - 0.40 = 0.60\)

Since passing both events is accounted for specifically with a given probability (\(0.12\)), this area where both fail must also take their joint success into account. Hence, Spike's probability of failing both chemistry and mathematics equals \((0.65 - 0.12) \times (0.60 - 0.12) = 0.3196\). This calculation involves looking beyond simple complements to ensure the true interrelationships of probabilities are respected.