Mathematics
Grade 12
15 min
Find probabilities using combinations and permutations
Find probabilities using combinations and permutations
Tutorial Preview
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Introduction & Learning Objectives
Learning Objectives
Set up a probability expression as a ratio of combinations or permutations for a variable number of items, n.
Formulate a limit expression for a probability as n approaches infinity.
Expand combination and permutation formulas into polynomial expressions in terms of n.
Evaluate limits of rational functions involving polynomials derived from combinatorics.
Apply leading term analysis to find the limiting probability of an event in an infinitely large sample space.
Interpret the meaning of a limiting probability (e.g., a limit of 0, 1, or a constant fraction).
Imagine a lottery with an infinite number of tickets and an infinite number of prizes. What are your chances of winning? 🎟️ Let's use calculus to find out!
This tutorial merges two key areas of...
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Key Concepts & Vocabulary
TermDefinitionExample
CombinationA selection of items from a collection, where the order of selection does not matter.Choosing 3 students from a group of 10 for a committee. The committee of {Ann, Bob, Cae} is the same as {Cae, Ann, Bob}.
PermutationAn arrangement of items from a collection, where the order of arrangement is important.Awarding gold, silver, and bronze medals to 3 runners from a group of 10. The arrangement of {Runner A, Runner B, Runner C} is different from {Runner B, Runner A, Runner C}.
Probability of an EventA measure of the likelihood of an event occurring, calculated as the ratio of favorable outcomes to the total number of possible outcomes.The probability of rolling a 4 on a standard six-sided die is 1 (favorable outcome) / 6 (total outcomes).
Limit at InfinityThe...
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Core Formulas
Probability Formula
P(A) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}
The fundamental formula for calculating the probability of an event A. In this context, the outcomes are counted using combinations or permutations.
Combination Formula
C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!}
Calculates the number of ways to choose k items from a set of n items where order does not matter. This expands to a polynomial in n.
Permutation Formula
P(n, k) = \frac{n!}{(n-k)!}
Calculates the number of ways to arrange k items from a set of n items where order matters. This also expands to a polynomial in n.
Limit of a Rational Function at Infinity
\lim_{n \to \infty} \frac{a_p n^p + ...}{b_q n^q + ...} = \begin{cases} 0 & p < q \\ \frac{a...
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Challenging
An urn contains n red, n blue, and n green marbles. If 3 marbles are drawn without replacement, what is the limit of the probability of drawing one marble of each color as n approaches infinity?
A.1/27
B.1/9
C.2/9
D.1/3
Challenging
The limit of the probability of an event as n approaches infinity is 1/8. Which of these scenarios matches this result?
A.Drawing 3 marbles from an urn with n red and n blue marbles, with the event being that all 3 are red.
B.Drawing 2 marbles from an urn with n red and 3n blue marbles, with the event being that both are red.
C.Arranging n people in a line, with the event that 3 specific friends are in the first 3 spots.
D.Drawing 3 marbles from an urn with n red and 2n blue marbles, with the event being that all 3 are red.
Challenging
You draw k marbles (where k is a fixed integer ≥ 2) from an urn containing n red and n blue marbles. What is the limit of the probability that all k marbles are red as n approaches infinity?
A.0
B.1/k!
C.1/2
D.1/2^k
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