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Probability

Conditional Probability

Intro

Concept Quizzes

Conditional Probability Warmup
Probability - Independent events
Two-Events Conditionals
Discrete Conditional Probabilities
Monty Hall
Conditional Probability Misconceptions
Maximizing Conditional Probability
Bayes' Theorem

Challenge Quizzes

Conditional Probability: Level 2 Challenges
Conditional Probability: Level 3 Challenges
Conditional Probability: Level 4 Challenges
Conditional Probability: Level 5 Challenges

Conditional Probability Misconceptions

Which of the following equalities is not necessarily true if $A, B$ are events?

P(A \cap B) = P(A) \cdot P(B)

P(A \cup B) = P(A) + P(B) - P(A \cap B)

P(A \cap B) = P(A) \cdot P(B \mid A)

P(A \cap B) = P(B) \cdot P(A \mid B)

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by Brilliant Staff

Suppose you flip a fair coin 10 times. What is the probability of the last two flips both being heads if you know that the first eight flips were heads?

More than

\frac{1}{4}

\frac{1}{4}

Less than

\frac{1}{4}

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by Brilliant Staff

$P(A \mid B) = P(B \mid A).$

which of the following must be true of $A$ and $B,$ if $P(A \cap B) > 0?$

They are independent

P(A \cap B) = P(A) \cdot P(B)

They are mutually exclusive They happen with equal probability

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by Brilliant Staff

Suppose there are two types of surgeries that doctors perform. Doctor A has a higher success rate than Doctor B on the first type of surgery. Doctor A also has a higher success rate than Doctor B on the second type of surgery. Is it true that Doctor A necessarily has a higher overall success rate than Doctor B?

It must be false Yes, it must be true It can be true, but not necessarily

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by Brilliant Staff

Assume that a child has an equal chance of being born on any month and of either gender (male, female). Let $p$ be the probability that given that a family has two children and at least one of them is a girl, both are girls. Furthermore, let $q$ be the probability that given that a family has two children and at least one of them is a girl born in April, that both are girls. Does $p = q$ ?

No Yes

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by Brilliant Staff