jesh

12/8/2025

Stats and Prob

Combination

Subjects

Stats and Prob

•

Dec 8, 2025

•

4 pages

Understanding Combinations

jesh

@tokwa_2ox3e

Ever wondered how to calculate your chances of getting a... Show more

1 / 4

$COMBINATION C(n,r) = \frac{n!}{r!(n-r)!} 1. How many diff. possible hands of 13 cards in a pack of 52 ordinary playing cards are there? 4$

Combination Formulas

Combinations help you figure out how many ways you can select items when order doesn't matter. The formula C(n,r) = n!/ $r!(n-r)!$ might look scary, but it's actually your best friend for solving selection problems.

Think about it this way - if you're choosing 13 cards from a deck of 52, there are over 635 billion different possible hands! That's why poker is so exciting. For smaller numbers, like selecting 3 candidates from 7 graduates, you get 35 different ways.

Here's the key trick to remember: when you see "and" in a problem, you multiply the combinations. When you see "or," you add them. So if mom needs to pick 3 arcade games AND 2 sports games, you multiply 10C3 × 5C2 = 1,200 ways.

Pro Tip: Start with smaller numbers to understand the pattern before tackling bigger combination problems!

$COMBINATION C(n,r) = \frac{n!}{r!(n-r)!} 1. How many diff. possible hands of 13 cards in a pack of 52 ordinary playing cards are there? 4$

Probability Basics

Probability is just a fancy way of asking "what are the chances?" You calculate it using P(E) = favorable outcomes/total outcomes, and the answer is always between 0 and 1.

There are two main methods you'll use. The empirical method uses real data - like when 24 out of 182 teenage drivers filed insurance claims, giving us a probability of about 0.1319 or 13.19%. The classical method uses theoretical equal chances.

For the classical method, think about flipping coins or rolling dice where each outcome is equally likely. When you want 3 children of the same gender, you have 2 favorable outcomes (all boys or all girls) out of 8 total possibilities, so P = 2/8 = 0.25.

Remember: All probabilities in a situation must add up to 1 - that's like saying "something has to happen!"

$COMBINATION C(n,r) = \frac{n!}{r!(n-r)!} 1. How many diff. possible hands of 13 cards in a pack of 52 ordinary playing cards are there? 4$

Dice Probability Examples

Rolling dice is one of the easiest ways to understand probability because each face has an equal chance of appearing. When you roll two fair dice, you have 36 total possible outcomes.

Getting a seven is pretty common - you can roll (1,6), (2,5), (3,4), (4,3), (5,2), or (6,1). That's 6 ways out of 36, so P = 6/36 ≈ 0.1667. But "snake eyes" (rolling two ones) only happens one way out of 36, making P = 1/36 ≈ 0.0278.

Things get interesting with loaded dice - dice that aren't fair. If even numbers are twice as likely as odd numbers, you need to adjust your sample space. Instead of each outcome being equally likely, even numbers get more weight in your calculations.

Watch Out: Always check if the problem mentions "fair" dice or cards - loaded/biased items change everything!

$COMBINATION C(n,r) = \frac{n!}{r!(n-r)!} 1. How many diff. possible hands of 13 cards in a pack of 52 ordinary playing cards are there? 4$

Classical vs Empirical Methods

Here's where theory meets reality! The empirical method uses actual survey data, while the classical method assumes perfect conditions. Both are useful, but they often give slightly different answers.

In a real survey of 300 families, 180 had two boys and one girl, giving us P = 180/300 = 0.36 using the empirical method. But if boys and girls were perfectly equally likely, the classical method gives us P = 3/8 = 0.375.

The small difference (0.36 vs 0.375) shows that real life doesn't always match theoretical predictions perfectly. Family planning, genetics, and social factors can create slight variations from the "ideal" probability.

Key Insight: Use empirical when you have real data, classical when you're dealing with theoretical "fair" situations like coins and dice!

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Subjects

Stats and Prob

•

Dec 8, 2025

•

4 pages

Understanding Combinations

jesh

@tokwa_2ox3e

Ever wondered how to calculate your chances of getting a perfect hand in cards or the probability of your favorite team winning? Combination and probability are mathematical tools that help us figure out how many ways things can happen and... Show more

$COMBINATION C(n,r) = \frac{n!}{r!(n-r)!} 1. How many diff. possible hands of 13 cards in a pack of 52 ordinary playing cards are there? 4$

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Combination Formulas

Pro Tip: Start with smaller numbers to understand the pattern before tackling bigger combination problems!

$COMBINATION C(n,r) = \frac{n!}{r!(n-r)!} 1. How many diff. possible hands of 13 cards in a pack of 52 ordinary playing cards are there? 4$

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Access to all documents

Improve your grades

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Probability Basics

Probability is just a fancy way of asking "what are the chances?" You calculate it using P(E) = favorable outcomes/total outcomes, and the answer is always between 0 and 1.

Remember: All probabilities in a situation must add up to 1 - that's like saying "something has to happen!"

$COMBINATION C(n,r) = \frac{n!}{r!(n-r)!} 1. How many diff. possible hands of 13 cards in a pack of 52 ordinary playing cards are there? 4$

Sign up to see the contentIt's free!

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Dice Probability Examples

Rolling dice is one of the easiest ways to understand probability because each face has an equal chance of appearing. When you roll two fair dice, you have 36 total possible outcomes.

Watch Out: Always check if the problem mentions "fair" dice or cards - loaded/biased items change everything!

$COMBINATION C(n,r) = \frac{n!}{r!(n-r)!} 1. How many diff. possible hands of 13 cards in a pack of 52 ordinary playing cards are there? 4$

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Access to all documents

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Classical vs Empirical Methods

Key Insight: Use empirical when you have real data, classical when you're dealing with theoretical "fair" situations like coins and dice!