Mathematics
Grade 10
15 min
Identify dependent and independent events
Identify dependent and independent events
What you'll learn
- Apply trigonometric ratios (sine, cosine, tangent) to solve for unknown side lengths in right-angled triangles with at least 80% accuracy in a problem set.
- Analyze a given word problem involving right triangles and accurately identify the appropriate trigonometric ratio to use for finding an unknown side length in at least 4 out of 5 scenarios.
- Solve real-world problems involving angles of elevation and depression by applying trigonometric ratios to calculate unknown distances or heights with correct units in a written exam.
- Explain the relationship between the angle of a right triangle and the corresponding trigonometric ratio used to determine a side length, justifying the choice of ratio in a written explanation.
Tutorial Preview
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Introduction & Learning Objectives
Learning Objectives
Define independent and dependent events using precise mathematical language.
Differentiate between scenarios involving sampling 'with replacement' and 'without replacement'.
Analyze a real-world scenario to determine if two or more events are dependent or independent.
Apply the formal multiplication rule, P(A and B) = P(A) * P(B), to test for independence.
Apply the conditional probability test, P(B|A) = P(B), to test for independence.
Explain why mutually exclusive events are always dependent.
You're playing a board game and need to roll a 6 to win. Does your previous roll of a 3 affect your chances of rolling a 6 now? 🎲 Let's investigate!
This tutorial will teach you how to distinguish between two fundamental types of eve...
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Key Concepts & Vocabulary
TermDefinitionExample
EventA specific outcome or a set of outcomes from a random experiment.In the experiment of rolling a six-sided die, an event could be 'rolling an even number', which corresponds to the set of outcomes {2, 4, 6}.
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' (Event A) and rolling a die and getting a '5' (Event B). The coin flip has no impact on the die roll.
Dependent EventsTwo events are dependent if the occurrence of one event changes the probability of the other event occurring.Drawing a King from a deck of cards (Event A), not replacing it, and then drawing another King (Event B). The first draw changes the co...
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Core Formulas
Test for Independence (Multiplication Rule)
P(A \cap B) = P(A) \times P(B)
Two events A and B are independent if and only if the probability that they both occur is equal to the product of their individual probabilities. This is the most common way to mathematically prove independence.
Test for Independence (Conditional Probability)
P(B|A) = P(B) \text{ or } P(A|B) = P(A)
Two events A and B are independent if and only if the probability of B occurring, given that A has already occurred, is the same as the original probability of B. This shows that A's occurrence had no effect on B.
Probability of Dependent Events
P(A \cap B) = P(A) \times P(B|A)
If two events are dependent, the probability of both occurring is the probability of the first event multiplied by the...
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Challenging
If two events, A and B, are mutually exclusive, and P(A) = 0.4 and P(B) = 0.3, which statement MUST be true?
A.The events are dependent because P(B|A) = 0, which is not equal to P(B).
B.The events are independent because P(A and B) = 0.
C.The events are independent because P(A) + P(B) is not 1.
D.The events are dependent because P(A) * P(B) is not 0.
Challenging
A box contains 10 processors, 3 of which are defective. You randomly select two processors without replacement. Let D1 be the event the first is defective, and D2 be the event the second is defective. Why are these events dependent?
A.The events are independent because the quality of one processor does not physically change the other.
B.The events are dependent because selecting a defective processor first changes the proportion of defective processors left in the box.
C.The events are independent because P(D1) = P(D2).
D.The events are dependent because P(D1 and D2) = P(D1) + P(D2).
Challenging
Events A and B are independent. If P(A) = 0.7 and P(B) = 0.2, what is the probability that either A or B (or both) will occur, i.e., P(A or B)?
A.0.90
B.0.14
C.0.76
D.0.50
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What grade level is "Identify dependent and independent events"?
Identify dependent and independent events is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Identify dependent and independent events?
You'll be able to: Apply trigonometric ratios (sine, cosine, tangent) to solve for unknown side lengths in right-angled triangles with at least 80% accuracy in a problem set; Analyze a given word problem involving right triangles and accurately….
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How many practice questions are included with Identify dependent and independent events?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.