Mathematics
Grade 11
15 min
Identify independent events
Identify independent events
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Introduction & Learning Objectives
Learning Objectives
Define statistical independence in the context of probability.
Differentiate between independent, dependent, and mutually exclusive events.
Apply the multiplication rule for independent events to test for independence.
Use conditional probability to formally test for independence.
Calculate the probability of the intersection of two events and compare it to the product of their individual probabilities.
Analyze scenarios, including those with contingency tables, to determine if events are independent.
If you flip a coin and it lands on heads, does that change the chance of your favorite team winning their next game? 🎲 Of course not! Let's explore the math behind this intuition.
This tutorial will teach you how to mathematically identify independent...
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Key Concepts & Vocabulary
TermDefinitionExample
EventA specific outcome or a set of outcomes from a random experiment.When rolling a standard six-sided die, 'rolling a 4' is an event. 'Rolling an even number' (the set {2, 4, 6}) is also an event.
Independent EventsTwo events are independent if the occurrence of one event does not affect the probability of the other event occurring.Flipping a coin and getting 'heads' is independent of rolling a die and getting a '5'. The coin's outcome has no influence on the die's outcome.
Dependent EventsTwo events are dependent if the occurrence of one event changes the probability of the other event occurring.Drawing a card from a deck and not replacing it, then drawing a second card. The probability of the second draw depends o...
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Core Formulas
Multiplication Rule for Independent Events
Two events A and B are independent if and only if: P(A \cap B) = P(A) \times P(B)
Use this formula to test for independence. Calculate the probability of both events happening together, P(A ∩ B). Then, calculate the product of their individual probabilities, P(A) × P(B). If these two values are equal, the events are independent.
Conditional Probability Test for Independence
Two events A and B (with P(B) > 0) are independent if and only if: P(A|B) = P(A)
This rule states that the probability of A happening, given that B has already happened, is the same as the original probability of A. This is the conceptual definition of independence. If knowing B happened doesn't change A's probability, they are independent.
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Challenging
Three events A, B, and C are mutually independent. If P(A) = 0.5, P(B) = 0.4, and P(C) = 0.2, what is the probability that none of these events occur?
A.0.04
B.0.1
C.0.9
D.0.24
Challenging
Let A and B be two events with P(A) > 0 and P(B) > 0. If A and B are mutually exclusive, which of the following provides the rigorous proof that they must be dependent?
A.P(A ∪ B) = P(A) + P(B), which is not the formula for independence.
B.P(A|B) = P(A ∩ B) / P(B) = 0 / P(B) = 0. Since P(A) > 0, P(A|B) ≠P(A).
C.P(A) + P(B) is not necessarily equal to 1.
D.The sample spaces for A and B do not overlap.
Challenging
In a class, the probability that a student has a laptop is P(L) = 0.7. The probability that a student has a tablet is P(T) = 0.5. The probability that a student has a tablet given they have a laptop is P(T|L) = 0.6. Are the events L and T independent, and what is the probability a student has both?
A.Dependent, because P(T|L) ≠P(T); P(L ∩ T) = 0.42
B.Independent, because P(T|L) > P(T); P(L ∩ T) = 0.35
C.Dependent, because P(L) > P(T); P(L ∩ T) = 0.30
D.Independent, because the problem involves technology; P(L ∩ T) = 0.35
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