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Hogar Banco de preguntas Matemáticas Probability Archive 2025 January 22

Probability Questions from Jan 22,2025

Browse the Probability Q&A Archive for Jan 22,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Exercice 22. Une urne contient 8 boules blanches et 2 boules noires. On tire sans remise et successivement 3 boules de cette urne. 1. Quelle est la probabilité qu'au moins une boule noire figure à l'intérieur du tirage? Exercice 2 Soient \( \left(S_{n}\right)_{n \geqslant 0} \) la marche aléatoire symétrique sur \( \mathbb{Z} \) et \( \left(\mathcal{F}_{n}\right)_{n \geqslant 0} \) sa filtration naturelle. 1. Montrer que \( \left(S_{n}\right) \) est une martingale. 2. Montrer que \( \left(S_{n}^{2}-n\right) \) est une martingale. 3. Soient \( a, b \geqslant 0 \) et \( T=\inf \left\{n \geqslant 0: S_{n}=-a\right. \) ou \( \left.S_{n}=b\right\} \). On suppose que \( T<+\infty \mathbb{P} \) p i) Montrer que \( \mathbb{P}\left(S_{T}=b\right)=\frac{a}{a+b} \) ii) Montrer que \( \mathbb{E}(T)=a b \) 14 A bag contains 5 blue marbles, 4 red marbles, and 1 yellow marble. The probability of drawing a blue marble out of the bag is \( \frac{5}{10} \) or \( 50 \% \). How many of what color of marbles must be added to the bag so that the probability of a blue marble being drawn at random from the bag is \( 90 \% \) ? Select the correct answer below and fill in the answer box within your choice. (Type a whole number.) A. There must be \( \square \) red marbles added to the bag. B. There must be \( \square \) yellow marbles added to the bag. C. There must be \( \square \) blue marbles added to the bag. What are dependent events in probability? The probabiling of randoml selecting a voter from he \( 17: 30-18: 00 \) time slot. How do you calculate the probability of rolling a total greater than 7 with two six-sided dice? How do you calculate the probability of rolling a total greater than 7 with two six-sided dice? How do you calculate the probability of rolling a total greater than 7 with two six-sided dice? How do you calculate the probability of rolling a total greater than 7 with two six-sided dice? Explain why choosing two socks from a drawer containing blue, red, and green socks is an example of dependent events.
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