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Home Question Bank Math 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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Question 7 \( P(A)=0.1 \) Compute the complement of \( \mathrm{P}(\mathrm{A}) \). If the risk of contracting a given illness at some point in one's life is 0.85 , what are the odds of contracting the illness at some point in one's life? 0.15 to 1 0.176 to 1 0.85 to 1 1.667 to 1 5.667 to 1 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? 7 amara is going to toss a number cube with faces labeled \( 2-7 \). She will record the result. Selec umbers that make up the sample space for Samara's experiment. A. 1 B. 2 C. 3 D. 4 E. 5 F. 6 G. 7 H. 8 How do you calculate the probability of rolling a total greater than 7 with two six-sided dice? Exercice 3. On considère une suite \( \left(X_{n}, n \geq 0\right) \) de v.a réelles indépendantes de normale \( N\left(m, \sigma^{2}\right) \) avec \( m<0 \). On pose \( S_{0}=0 \) et \[ S_{n}=X_{0}+, \ldots+X_{n} \text { et } \mathcal{F}_{n}=\sigma\left(S_{0}, \ldots, S_{n}\right) \] Soit la v.a \( W=\sup _{n \geq 0} S_{n} \) 1. Montrer que \( P(W<+\infty)=1 \) 2. Calculer \( E\left(e^{\left.\lambda S_{n+1} \mid \mathcal{F}_{n}\right) \text { pour tout } \lambda \in \mathbb{R} \text {. }}\right. \) 3. Montrer qu'il existe un unique \( \lambda_{0}>0 \) tel que \( \left(e^{\lambda_{0} S_{n}}, n \geq 0\right) \) soit une martingale 4. Montrer que, pour tout \( \alpha>0 \), on a \( P\left(e^{\lambda_{0} W}>\alpha\right) \leq \frac{1}{\alpha} \) 5. En déduire que pour tout \( t>0 \), on a \( P(W>t) \leq e^{-\lambda_{0} t} \) 6. Montrer que \( E\left(e^{\lambda W}\right)=1+\lambda \int_{0}^{+\infty} e^{\lambda t} P(W>t) d t \) 7. En déduire que pour tout \( \lambda<\lambda_{0} \), on a \( E\left(e^{\lambda W}\right)<+\infty \) Soit la v.a \( W=\sup _{n \geq 0} S_{n} \) 1. Montrer que \( P(W<+\infty)=1 \) 2. Calculer \( E\left(e^{\lambda S_{n+1}} \mid \mathcal{F}_{n}\right) \) pour tout \( \lambda \in \mathbb{R} \). 3. Montrer qu'il existe un unique \( \lambda_{0}>0 \) tel que \( \left(e^{\lambda_{0} S_{n}}, n \geq 0\right) \) soit une martingale 4. Montrer que, pour tout \( \alpha>0 \), on a \( P\left(e^{\lambda_{0} W}>\alpha\right) \leq \frac{1}{\alpha} \) 5. En déduire que pour tout \( t>0 \), on a \( P(W>t) \leq e^{-\lambda_{0} t} \) 6. Montrer que \( E\left(e^{\lambda W}\right)=1+\lambda \int_{0}^{+\infty} e^{\lambda t} P(W>t) d t \) 7. En déduire que pour tout \( \lambda<\lambda_{0} \), on a \( E\left(e^{\lambda W}\right)<+\infty \) A deck of playing cards has four suits, with thirteen cards Which statements are true? Check all that apply. in each suit consisting of the numbers 2 through 10, a jack, a queen, a king, and an ace. The four suits are hearts, diamonds, spades, and clubs. A hand of five cards will be chosen at random. The total possible outcomes can be found using \( { }_{52} \mathrm{C}_{5} \). The total possible outcomes can be found using \( { }_{52} \mathrm{P}_{5} \). The probability of choosing two diamonds and three hearts is 0.089 . The probability of choosing five spades is roughly 0.05 \[ P(X=x)=\left\{\begin{array}{lll}T x & \text { for } & x=1,3,5 \\ T(x-3) & \text { for } & x=4 \\ 0 & \text { elsewhere }\end{array}\right. \] where \( T \) is a constant. a) Find the value of \( T \).
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