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Omni-MATH / Brave NiuNiu (a milk drink company) organizes a promotion during the Chinese New…

Problem

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problem

Brave NiuNiu (a milk drink company) organizes a promotion during the Chinese New Year: one gets a red packet when buying a carton of milk of their brand, and there is one of the following characters in the red packet "虎"(Tiger), "生"(Gain), "威"(Strength). If one collects two "虎", one "生" and one "威", then they form a Chinese phrases "虎虎生威" (Pronunciation: hu hu sheng wei), which means "Have the courage and strength of the tiger". This is a nice blessing because the Chinese zodiac sign for the year 2022 is tiger. Soon, the product of Brave NiuNiu becomes quite popular and people hope to get a collection of "虎虎生威". Suppose that the characters in every packet are independently random, and each character has probability 13\frac{1}{3}. What is the expectation of cartons of milk to collect "虎虎生威" (i.e. one collects at least 2 copies of "虎", 1 copy of "生", 1 copy of "威")? Options: (A) 6136 \frac{1}{3}, (B) 7137 \frac{1}{3}, (C) 8138 \frac{1}{3}, (D) 9139 \frac{1}{3}, (E) None of the above.
Plain-text mathematical notation (without MathML)
Brave NiuNiu (a milk drink company) organizes a promotion during the Chinese New Year: one gets a red packet when buying a carton of milk of their brand, and there is one of the following characters in the red packet "虎"(Tiger), "生"(Gain), "威"(Strength). If one collects two "虎", one "生" and one "威", then they form a Chinese phrases "虎虎生威" (Pronunciation: hu hu sheng wei), which means "Have the courage and strength of the tiger". This is a nice blessing because the Chinese zodiac sign for the year 2022 is tiger. Soon, the product of Brave NiuNiu becomes quite popular and people hope to get a collection of "虎虎生威". Suppose that the characters in every packet are independently random, and each character has probability (1)/(3). What is the expectation of cartons of milk to collect "虎虎生威" (i.e. one collects at least 2 copies of "虎", 1 copy of "生", 1 copy of "威")? Options: (A) 6(1)/(3), (B) 7(1)/(3), (C) 8(1)/(3), (D) 9(1)/(3), (E) None of the above.
Original LaTeX notation
Brave NiuNiu (a milk drink company) organizes a promotion during the Chinese New Year: one gets a red packet when buying a carton of milk of their brand, and there is one of the following characters in the red packet "虎"(Tiger), "生"(Gain), "威"(Strength). If one collects two "虎", one "生" and one "威", then they form a Chinese phrases "虎虎生威" (Pronunciation: hu hu sheng wei), which means "Have the courage and strength of the tiger". This is a nice blessing because the Chinese zodiac sign for the year 2022 is tiger. Soon, the product of Brave NiuNiu becomes quite popular and people hope to get a collection of "虎虎生威". Suppose that the characters in every packet are independently random, and each character has probability $\frac{1}{3}$. What is the expectation of cartons of milk to collect "虎虎生威" (i.e. one collects at least 2 copies of "虎", 1 copy of "生", 1 copy of "威")? Options: (A) $6 \frac{1}{3}$, (B) $7 \frac{1}{3}$, (C) $8 \frac{1}{3}$, (D) $9 \frac{1}{3}$, (E) None of the above.

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Source and history

task_id
a7cf9ce5-7623-5d1f-ab25-eefff80c9846
task_key
test--a7cf9ce5-7623-5d1f-ab25-eefff80c9846
task_revision_id
1
config / split
/ test

Official source

initial import