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"Two Children Puzzle / Boy Born on a Tuesday" and variants:

• In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
• New Two Children problem
• Boy Born on a Tuesday - is it just a language trick?

A "liar paradox" variant: Multiple-choice question about the probability of a random answer to itself being correct

Bertrand's box paradox

Expectation as tail probability (complementary CDF) integral/sum.

• Continuous (generalized to any $E[X^p]$ in the answer): Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$.

• Discrete: Find the Mean for Non-Negative Integer-Valued Random Variable

• Measure-theoretic treatment: Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$

Probability for roots of a quadratic equation to be real, where the coefficients are iid random variables with distribution:

discrete uniform$\{1,2, 3,4,5,6\}$ from standard die, whereas 322939 is also a good candidate. ^{There are isolated posts regarding discrete uniform of different ranges, and continuous uniform, exponential distribution, etc.}

Coupon collector's problem coupon-collector

• standard version
  • expectation: Expected time to roll all $1$ through $6$ on a die (duplicated at calculating expected number of packets.)
  • probability distribution: Probability distribution in the coupon collector's problem
• only desired items
  • expectation: Coupon collector problem with partial collection of a specific set of coupons (duplicated at Pile of cards - Probability question)
  • probability distribution: Coupon Collector Prob Variation (duplicated at Coupon collector - Chance of collecting desired coupons but not a full set.)
• batches / groups / packets
  • fixed number per packet, distinct coupons:
    • expectation: Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen (duplicated at Probability - Expected number of draws to get all 52 cards at least once drawing in groups of size n and Expected number of games so that every player has been an Imposter in Among Us)
    • probability distribution: Sample all elements from a set at least once, with replacement
  • fixed number per packet, repetitions allowed:
    • expectation: Coupon Collector Problem with Batched Selections (duplicated at How to calculate the expected value of the coupon collector problem if we are collecting the coupons in groups of k?)
    • probability distribution: Coupon Collector's problem, version with multiple coupons in a box
  • random number per packet, distinct coupons: coupon collector problem with groups
• unequal probabilities:
  • mean and variance: Coupon collector's problem: mean and variance in number of coupons to be collected to complete a set (unequal probabilities)
  • probability distribution: Probability that N i.i.d. draws from a multinomial distribution have made all events appear

"Two Children Puzzle / Boy Born on a Tuesday" and variants:

• In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
• New Two Children problem
• Boy Born on a Tuesday - is it just a language trick?

A "liar paradox" variant: Multiple-choice question about the probability of a random answer to itself being correct

Bertrand's box paradox

Expectation as tail probability (complementary CDF) integral/sum.

• Continuous (generalized to any $E[X^p]$ in the answer): Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$.

• Discrete: Find the Mean for Non-Negative Integer-Valued Random Variable

• Measure-theoretic treatment: Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$

Probability for roots of a quadratic equation to be real, where the coefficients are iid random variables with distribution:

• discrete uniform$\{1,2, 3,4,5,6\}$ from standard die, whereas 322939 is also a good candidate. ^{There are isolated posts regarding discrete uniform of different ranges, and continuous uniform, exponential distribution, etc.}

Coupon collector's problem coupon-collector

• standard version
  • expectation: Expected time to roll all $1$ through $6$ on a die (duplicated at calculating expected number of packets.)
  • probability distribution: Probability distribution in the coupon collector's problem
• only desired items
  • expectation: Coupon collector problem with partial collection of a specific set of coupons (duplicated at Pile of cards - Probability question)
  • probability distribution: Coupon Collector Prob Variation (duplicated at Coupon collector - Chance of collecting desired coupons but not a full set.)
• batches / groups / packets
  • fixed number per packet, distinct coupons:
    • expectation: Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen (duplicated at Probability - Expected number of draws to get all 52 cards at least once drawing in groups of size n and Expected number of games so that every player has been an Imposter in Among Us)
    • probability distribution: Sample all elements from a set at least once, with replacement
  • fixed number per packet, repetitions allowed:
    • expectation: Coupon Collector Problem with Batched Selections (duplicated at How to calculate the expected value of the coupon collector problem if we are collecting the coupons in groups of k?)
    • probability distribution: Coupon Collector's problem, version with multiple coupons in a box
  • random number per packet, distinct coupons: coupon collector problem with groups
• unequal probabilities:
  • mean and variance: Coupon collector's problem: mean and variance in number of coupons to be collected to complete a set (unequal probabilities)
  • probability distribution: Probability that N i.i.d. draws from a multinomial distribution have made all events appear

"Two Children Puzzle / Boy Born on a Tuesday" and variants:

• In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
• New Two Children problem
• Boy Born on a Tuesday - is it just a language trick?

A "liar paradox" variant: Multiple-choice question about the probability of a random answer to itself being correct

Bertrand's box paradox

Expectation as tail probability (complementary CDF) integral/sum.

• Continuous (generalized to any $E[X^p]$ in the answer): Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$.

• Discrete: Find the Mean for Non-Negative Integer-Valued Random Variable

• Measure-theoretic treatment: Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$

Probability for roots of a quadratic equation to be real, where the coefficients are iid random variables with distribution:

discrete uniform$\{1,2, 3,4,5,6\}$ from standard die, whereas 322939 is also a good candidate. ^{There are isolated posts regarding discrete uniform of different ranges, and continuous uniform, exponential distribution, etc.}

Coupon collector's problem coupon-collector

• standard version
  • expectation: Expected time to roll all $1$ through $6$ on a die (duplicated at calculating expected number of packets.)
  • probability distribution: Probability distribution in the coupon collector's problem
• only desired items
  • expectation: Coupon collector problem with partial collection of a specific set of coupons (duplicated at Pile of cards - Probability question)
  • probability distribution: Coupon Collector Prob Variation (duplicated at Coupon collector - Chance of collecting desired coupons but not a full set.)
• batches / groups / packets
  • fixed number per packet, distinct coupons:
    • expectation: Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen (duplicated at Probability - Expected number of draws to get all 52 cards at least once drawing in groups of size n)
    • probability distribution: Sample all elements from a set at least once, with replacement
  • fixed number per packet, repetitions allowed:
    • expectation: Coupon Collector Problem with Batched Selections (duplicated at How to calculate the expected value of the coupon collector problem if we are collecting the coupons in groups of k?)
    • probability distribution: Coupon Collector's problem, version with multiple coupons in a box
  • random number per packet, distinct coupons: coupon collector problem with groups
• unequal probabilities:
  • mean and variance: Coupon collector's problem: mean and variance in number of coupons to be collected to complete a set (unequal probabilities)
  • probability distribution: Probability that N i.i.d. draws from a multinomial distribution have made all events appear

"Two Children Puzzle / Boy Born on a Tuesday" and variants:

• In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
• New Two Children problem
• Boy Born on a Tuesday - is it just a language trick?

A "liar paradox" variant: Multiple-choice question about the probability of a random answer to itself being correct

Bertrand's box paradox

Expectation as tail probability (complementary CDF) integral/sum.

• Continuous (generalized to any $E[X^p]$ in the answer): Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$.

• Discrete: Find the Mean for Non-Negative Integer-Valued Random Variable

• Measure-theoretic treatment: Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$

Probability for roots of a quadratic equation to be real, where the coefficients are iid random variables with distribution:

discrete uniform$\{1,2, 3,4,5,6\}$ from standard die, whereas 322939 is also a good candidate. ^{There are isolated posts regarding discrete uniform of different ranges, and continuous uniform, exponential distribution, etc.}

Coupon collector's problem coupon-collector

• standard version
  • expectation: Expected time to roll all $1$ through $6$ on a die (duplicated at calculating expected number of packets.)
  • probability distribution: Probability distribution in the coupon collector's problem
• only desired items
  • expectation: Coupon collector problem with partial collection of a specific set of coupons (duplicated at Pile of cards - Probability question)
  • probability distribution: Coupon Collector Prob Variation (duplicated at Coupon collector - Chance of collecting desired coupons but not a full set.)
• batches / groups / packets
  • fixed number per packet, distinct coupons:
    • expectation: Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen (duplicated at Probability - Expected number of draws to get all 52 cards at least once drawing in groups of size n and Expected number of games so that every player has been an Imposter in Among Us)
    • probability distribution: Sample all elements from a set at least once, with replacement
  • fixed number per packet, repetitions allowed:
    • expectation: Coupon Collector Problem with Batched Selections (duplicated at How to calculate the expected value of the coupon collector problem if we are collecting the coupons in groups of k?)
    • probability distribution: Coupon Collector's problem, version with multiple coupons in a box
  • random number per packet, distinct coupons: coupon collector problem with groups
• unequal probabilities:
  • mean and variance: Coupon collector's problem: mean and variance in number of coupons to be collected to complete a set (unequal probabilities)
  • probability distribution: Probability that N i.i.d. draws from a multinomial distribution have made all events appear

"Two Children Puzzle / Boy Born on a Tuesday" and variants:

• In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
• New Two Children problem
• Boy Born on a Tuesday - is it just a language trick?

A "liar paradox" variant: Multiple-choice question about the probability of a random answer to itself being correct

A bag that contains a coin with two heads, a coin with two tails, and a standard coin; one is brought out and flipped; it shows heads. What is the probability that it is the standard coin?

• Bertrand's box paradox
• Card probability problem
• A probability problem?
• Statistics: Bertrand's Box Paradox

Expectation as tail probability (complementary CDF) integral/sum.

• Continuous (generalized to any $E[X^p]$ in the answer): Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$.

• Discrete: Find the Mean for Non-Negative Integer-Valued Random Variable

• Measure-theoretic treatment: Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$

Probability for roots of a quadratic equation to be real, where the coefficients are iid random variables with distribution:

discrete uniform$\{1,2, 3,4,5,6\}$ from standard die, whereas 322939 is also a good candidate. ^{There are isolated posts regarding discrete uniform of different ranges, and continuous uniform, exponential distribution, etc.}

Coupon collector's problem coupon-collector

• standard version
  • expectation: Expected time to roll all $1$ through $6$ on a die (duplicated at calculating expected number of packets.)
  • probability distribution: Probability distribution in the coupon collector's problem
• only desired items
  • expectation: Coupon collector problem with partial collection of a specific set of coupons (duplicated at Pile of cards - Probability question)
  • probability distribution: Coupon Collector Prob Variation (duplicated at Coupon collector - Chance of collecting desired coupons but not a full set.)
• batches / groups / packets
  • fixed number per packet, distinct coupons:
    • expectation: Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen (duplicated at Probability - Expected number of draws to get all 52 cards at least once drawing in groups of size n)
    • probability distribution: Sample all elements from a set at least once, with replacement
  • fixed number per packet, repetitions allowed:
    • expectation: Coupon Collector Problem with Batched Selections (duplicated at How to calculate the expected value of the coupon collector problem if we are collecting the coupons in groups of k?)
    • probability distribution: Coupon Collector's problem, version with multiple coupons in a box
  • random number per packet, distinct coupons: coupon collector problem with groups
• unequal probabilities:
  • mean and variance: Coupon collector's problem: mean and variance in number of coupons to be collected to complete a set (unequal probabilities)
  • probability distribution: Probability that N i.i.d. draws from a multinomial distribution have made all events appear

"Two Children Puzzle / Boy Born on a Tuesday" and variants:

• In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?
• New Two Children problem
• Boy Born on a Tuesday - is it just a language trick?

A "liar paradox" variant: Multiple-choice question about the probability of a random answer to itself being correct

Bertrand's box paradox

Expectation as tail probability (complementary CDF) integral/sum.

• Continuous (generalized to any $E[X^p]$ in the answer): Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$.

• Discrete: Find the Mean for Non-Negative Integer-Valued Random Variable

• Measure-theoretic treatment: Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$

Probability for roots of a quadratic equation to be real, where the coefficients are iid random variables with distribution:

discrete uniform$\{1,2, 3,4,5,6\}$ from standard die, whereas 322939 is also a good candidate. ^{There are isolated posts regarding discrete uniform of different ranges, and continuous uniform, exponential distribution, etc.}

Coupon collector's problem coupon-collector

• standard version
  • expectation: Expected time to roll all $1$ through $6$ on a die (duplicated at calculating expected number of packets.)
  • probability distribution: Probability distribution in the coupon collector's problem
• only desired items
  • expectation: Coupon collector problem with partial collection of a specific set of coupons (duplicated at Pile of cards - Probability question)
  • probability distribution: Coupon Collector Prob Variation (duplicated at Coupon collector - Chance of collecting desired coupons but not a full set.)
• batches / groups / packets
  • fixed number per packet, distinct coupons:
    • expectation: Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen (duplicated at Probability - Expected number of draws to get all 52 cards at least once drawing in groups of size n)
    • probability distribution: Sample all elements from a set at least once, with replacement
  • fixed number per packet, repetitions allowed:
    • expectation: Coupon Collector Problem with Batched Selections (duplicated at How to calculate the expected value of the coupon collector problem if we are collecting the coupons in groups of k?)
    • probability distribution: Coupon Collector's problem, version with multiple coupons in a box
  • random number per packet, distinct coupons: coupon collector problem with groups
• unequal probabilities:
  • mean and variance: Coupon collector's problem: mean and variance in number of coupons to be collected to complete a set (unequal probabilities)
  • probability distribution: Probability that N i.i.d. draws from a multinomial distribution have made all events appear