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Probability is the measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. In the context of the exercise, we are interested in finding the probability that a specific event, selecting a contaminated can, happens when we choose 3 cans out of 24.

To calculate this probability, we take the ratio of favorable outcomes to the total possible outcomes. Here, the favorable outcomes are the scenarios where our selected 3 cans include the contaminated can. The total possible outcomes are all the ways we can choose 3 cans from 24, which forms the denominator in our probability expression. The probability formula used is:

• \( P( ext{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \)
• In our example, the probability of choosing the contaminated can equals \( \frac{253}{2024} \).
• This simplifies to approximately 0.125 or 12.5%, indicating a fairly low chance.

The steps involved in calculating this always focus first on identifying the favorable outcomes relative to the overall possible scenarios, allowing us to express the probability as a fraction of those values.

Combinations deal with the selection of items from a larger set where the order does not matter. They are a key concept in combinatorics, which is the field of mathematics focused on counting and arranging possibilities.

• In our exercise, we are asked to identify how many combinations of 3 cans can be selected from 24. This involves using the combination formula: \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]
• Here, \( n = 24 \) and \( r = 3 \), so the calculation is: \[\binom{24}{3} = \frac{24!}{3! \times 21!} = 2024\]
• The result indicates there are 2024 different ways to choose any 3 cans from a group of 24.

Combinatorial counting like this is crucial for calculating probabilities, as it tells us the total number of outcomes possible. This, in turn, helps us identify the subset of favorable outcomes needed to determine probabilities.

The process of selecting the contaminated can can be broken down into specific logical steps.

• First, we treat selecting the contaminated can as a separate initial step. We only have one choice to include this specific can, hence there's 1 way to select it.
• Next, we need to choose the remaining 2 cans from the other 23. This is where combinations come in handy again: \[\binom{23}{2} = \frac{23 \times 22}{2 \times 1} = 253\]
• These 253 combinations represent all the possible ways we can form a group of 3 cans that includes the contaminated can.

The idea of contamination emphasizes the importance of identifying specific items within a larger set. Recognizing which selections involve the contaminated item is crucial to solving problems that require targeting specific outcomes within all possible scenarios. By breaking down the selection process into smaller, logical components, we can better calculate and understand specific probabilities tied to these targeted selections.