🎲 probability

1. **Problem Statement:** Three girls Jane, Alice, and Maxine complete a crossword puzzle. The probabilities of each getting it correct are Jane: $\frac{2}{5}$, Alice: $\frac{2}{3}

1. **Problem statement:** Given a Poisson random variable $X$ such that $P[X=2] = \frac{2}{3} P[X=1]$, find $P[X=3]$. 2. **Recall the Poisson probability mass function (pmf):**

1. The problem is to find the probability of the union of two events $A$ and $B$, given by the formula: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

1. **State the problem:** We want to find the probability of getting at least 4 heads when a fair coin is tossed 5 times. 2. **Formula and rules:** The number of heads in 5 tosses

1. **State the problem:** We want to find the probability that it will rain on at most one day during the 7 days the Hiking Club is camping, given the probability of rain on any da

1. **Problem statement:** Given a Poisson random variable $X$ with $P(X=0) = \frac{1}{2}$, find the expected value $E(X)$. 2. **Recall the Poisson distribution formula:**

1. Problem: Find $z$ such that $P(X > z + \mu_x) = \frac{1}{4}$ for $X \sim \text{Uniform}(1,2)$.\nStep 1: Mean of uniform $\mu_x = \frac{1+2}{2} = 1.5$.\nStep 2: We want $P(X > z

1. **Problem Statement:** Find the probability of selecting exactly 2 dark chocolates. 2. **Formula:** The probability of exactly $k$ successes in $n$ trials in a binomial distribu

1. **Problem statement:** Find the probability that exactly two dark chocolates are selected from a sequence of three chocolates drawn without replacement.

1. ଏହି ସମସ୍ୟାରେ, ଆମେ ଏକ 6-ପାର୍ଶ୍ୱ ଲୁଡୁ ଗୋଟି ପକାଇଛୁ ଓ ପ୍ରଶ୍ନ କରାଯାଇଛି \(P(5 \text{ ନୁହେଁ})\) କେତେ, ଯାହାର ଅର୍ଥ ହେଉଛି 5 ନ ଆସିବାର ସମ୍ଭାବନା। 2. ଏକ 6-ପାର୍ଶ୍ୱ ଲୁଡୁରେ ସମସ୍ତ ପାର୍ଶ୍ୱର ସମ୍ଭାବ

1. **Problem statement:** Given a Poisson random variable $X$ such that $P(X=0) = P(X=1)$, find the expected value $E(X)$. 2. **Recall the Poisson distribution formula:**

1. **Problem:** Let $X$ be binomial with $n=25$, $p=0.2$. Find $P(X < \mu_x - 2\sigma_x)$. 2. **Formulas:** For binomial, $\mu_x = np$, $\sigma_x = \sqrt{np(1-p)}$.

1. **Problem statement:** Find the probability that an integer chosen between 70 and 100 is (i) a prime number, (ii) divisible by 7.

1. **State the problem:** We want to find the probability that a randomly chosen pasty is vegetarian. 2. **Understand the data:** From the frequency tree:

1. **State the problem:** We have three gemstones worth 10, 100, and 1000 respectively. We draw two without replacement and want to find the expected value of the total value $H$ o

1. **Problem Statement:** Two fair dice are rolled simultaneously. Find the probability that: 1. The sum of the numbers is an even number greater than 7.

1. **Problem statement:** We are given probabilities related to the election of two members, Avantika and Jyotsna, and we need to find the probability that both are not elected tog

1. **Stating the problem:** We are given probabilities related to Raj and Rohan loving Simran. - Probability Raj loves Simran: $P(R) = 0.7$

1. **Stating the problem:** We have 7 candies of different flavors: orange, pineapple, mango, raspberry, and black currant. We want to find the probability of getting all black cur

1. **Problem Statement:** Two fair dice are rolled simultaneously. We need to find: - The probability that the sum of the numbers is an even number greater than 7.

1. The problem is to find the probability of getting exactly 2 tails when a fair coin is tossed three times. 2. The total number of possible outcomes when tossing a coin three time