Probability and Counting

Discrete Math · Axiom Academy

LESSON Probability and Counting Discover how counting outcomes connects directly to calculating probabilities, from simple coin flips to the binomial distribution. When all outcomes have the same chance of occurring, probability calculation becomes straightforward: just count! 2. The Multiplication Principle When we flip multiple coins or roll multiple dice, how many total outcomes are there? The multiplication principle tells us to multiply the number of choices at each step. What if we want to know: "In 5 coin flips, what's the probability of getting exactly 3 heads?" We need to count how many of the 2 5 = 32 total outcomes have exactly 3 heads. This is where combinations come in. We're choosing which 3 positions (out of 5) will be heads. The order doesn't matter for counting—HHHTT is the same outcome as HTHTH in terms of "3 heads total." Putting it all together: For n independent trials (like coin flips), each with probability p of success, the probability of exactly k successes is: C(n,k) : Number of ways to get k successes (counting) p k : Probability of k successes (1-p) n-k : Probability of n-k failures

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