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In probability theory, cumulative probability refers to the probability that a random variable will have a value less than or equal to a specific value. It helps in determining the likelihood of a variable falling within a certain range or below a specific cutoff.
In the context of our fishing exercise, we are dealing with 24 strikes and a 44% chance of catching a fish for each strike. To find the cumulative probability for catching 12 or fewer fish, we need to add up the probabilities of catching from 0 to 12 fish in total. You often use statistical software or a cumulative probability table to simplify this task because it involves calculating the probability of 13 different outcomes.
This total probability from 0 to 12 represents the cumulative probability, denoted as \( P(X \leq 12) \). By calculating this cumulative value, it allows us to know the likelihood of catching at most 12 fish.

Complementary probability revolves around the concept that the sum of the probabilities of all possible outcomes in a discrete probability scenario is equal to 1. When you know the likelihood of an event happening, its complementary probability is simply one minus that likelihood.
For the fishing problem where we want the probability of catching 5 or more fish, the complement helps simplify calculations. Rather than adding up all probabilities of catching from 5 to 24 fish, we first calculate the probability of catching fewer than 5 fish and subtract that from 1.
This logic uses the idea that knowing one portion of all possible outcomes ("fewer than 5") helps us find the remaining portion ("5 or more"). It's often much easier to calculate, particularly when dealing with a large number of possible outcomes.

Probability calculations in a binomial distribution rely on a specific formula that makes use of combinations and the given success probability. For our fishing exercise, the basic unit of calculation involves finding the probability of a specific number of catches (say exactly 7 fish out of 24 attempts).
To generalize this into finding ranges of probabilities, such as catching between a range of fish, we can apply cumulative probabilities and logical deductions with concepts like complementary probability.

• The probability function for a binomial distribution is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where:
• \( n \) is the total number of trials (24 strikes).
• \( k \) is the exact number of successes (fish caught).
• \( p \) is the probability of success on a single trial (44% or 0.44).
• Applying this formula to specific ranges, like finding the cumulative probability from 5 to 12, involves evaluating this expression at multiple points (5, 6, 7, ..., 12) and adding them together.

This comprehensive approach allows us to compute necessary probabilities for any specified range or number of outcomes in a binomially distributed scenario.