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Probability calculation is a crucial concept in understanding random events and their outcomes. It gives us the chance, or likelihood, of occurrence for a specific event.
This is often expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
When we're dealing with situations like a baseball game, we apply probability calculations to determine the likelihood of various outcomes.
This requires us to know two primary things:

• The number of trials, which are the occasions when an action is repeated, and in our case, the number of times Albert Pujols batted, which is 3 times.
• The chance of success, which here is getting a hit, with a probability of 0.359 each time he bats.

Using these, we can calculate the probability of different outcomes:
such as getting exactly three hits, none at all, or at least one within a given number of trials.

Combinatorics is the branch of mathematics dealing with counting combinations and permutations. In probability, it helps us understand how outcomes can be combined in experiments.
Each experiment follows specific rules to determine how the results can occur.
For instance, when you determine the probability of getting three hits in a row, you use the binomial coefficient from combinatorics.

• This is represented as \( \binom{n}{k} \), where \( n \) is the number of trials (like the number of at-bats) and \( k \) is the number of desired successes (in this case, the number of hits).

The binomial coefficient tells us how many ways \( k \) successful events can occur in \( n \) trials.
For example, \( \binom{3}{3} \) in our baseball scenario indicates there is only 1 way to achieve all hits in three attempts.
Understanding combinatorics allows you to properly calculate probabilities by accounting for all potential successful outcomes.

Complementary probability is a useful shortcut in many calculations by focusing on the event's complement, or what you don't want to occur. Instead of calculating the probability of at least one hit by adding up probabilities of one, two, or three hits, you can calculate the probability of zero hits and subtract it from 1.
This is because getting at least one hit means not getting zero hits.

• Mathematically, it's expressed as \( P(X \geq 1) = 1 - P(X = 0) \).

By using complementary probability, it simplifies the process and reduces calculation errors.
In Albert Pujols's example, by knowing the probability of zero hits (0.263), we quickly find the probability of getting at least one hit by subtracting this from 1, resulting in a probability of 0.737.
This method is particularly effective in simplifying calculations where computing individual event probabilities manually would be cumbersome.