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Probability is a mathematical concept that measures the likelihood of an event occurring. It's expressed as a number between 0 and 1. A probability of 0 means an event will not happen, while a probability of 1 indicates certainty it will happen. For example, saying the probability of the sun rising tomorrow is 1 means we are certain it will occur.

Understanding probability is crucial in making predictions and decisions based on uncertain outcomes. In our everyday lives, it helps us weigh risks and make informed choices. Mathematically, probability can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

• A probability value of 0.50 implies there's an equal chance of the event happening or not happening.
• The higher the probability, the more likely the event is.
• Probability can be measured as a fraction, decimal, or percentage.

Events are called independent if the occurrence of one event does not affect the probability of the other event happening. In simpler terms, they do not "influence" each other. If you flip a coin and roll a dice, the coin landing on heads does not influence what number you roll.

In the exercise example, the probability of a power failure and the probability of the generator stopping are independent. This means that knowing the power failed doesn't change the chance the generator will fail, and vice versa. This independence must be considered when calculating the probability that both events happen together.

The probability multiplication rule helps us find the probability of two independent events happening at the same time. If you have two independent events, A and B, the rule states that their combined probability is found by multiplying their individual probabilities.

In math terms, if the probability of event A happening is \( P(A) \) and the probability of event B happening is \( P(B) \), then the combined probability of both A and B happening is \( P(A) \times P(B) \).

For the exercise, the probability of a power failure \( (0.30) \) and the probability of the generator stopping \( (0.09) \) are independent, so their combined probability is found by multiplying these two values. Therefore, \( 0.30 \times 0.09 = 0.027 \).

Combined probability refers to the likelihood of multiple events happening together. In particular, when dealing with independent events, we simply multiply their individual probabilities to find the combined probability, as outlined by the multiplication rule. This tells us the chance that both events occur simultaneously in a given situation.

In our exercise, calculating the combined probability of losing both power sources during a snowstorm means seeing how both events (power failure and generator failure) overlap in occurrence. It's important in developing strategies, such as contingency planning, to reckon risk reliably when independent factors are involved. Using the math, a 2.7% chance was calculated for both failures occurring, informing decisions about possible upgrades or backups.