### Example Problem **Problem:** Pizza Shack usually seats 4.2 customers every 10 minutes during the lunch service. What is the likelihood it takes between 2 and 4 minutes for the next customer to arrive? **Explanation:** In this problem, we need to figure out the probability that the time until the next customer arrives falls between 2 and 4 minutes. Given the rate at which customers arrive (4.2 per 10 minutes), this kind of problem can be approached using the concepts of Poisson processes and exponential distributions. 1. **Average Rate of Customer Arrivals (λ):** - The average rate λ can be calculated as 4.2 customers per 10 minutes or 0.42 customers per minute. 2. **Exponential Distribution Function:** - For a continuous variable, the probability density function (PDF) for the exponential distribution is given by: \[ f(t) = λe^{-λt} \] where \( t \) is the time in minutes. 3. **Calculate the Probability:** - To determine the probability that the time \( t \) is between 2 and 4 minutes, we need to compute the cumulative distribution function (CDF) for these time points and find the difference: \[ P(2 \leq t \leq 4) = F(4) - F(2) \] where \( F(t) = 1 - e^{-λt} \). 4. **Applying the Values:** - Compute \( F(4) \) and \( F(2) \): \[ F(4) = 1 - e^{-0.42 \times 4} = 1 - e^{-1.68} \] \[ F(2) = 1 - e^{-0.42 \times 2} = 1 - e^{-0.84} \] - Now find the difference: \[ P(2 \leq t \leq 4) = (1 - e^{-1.68}) - (1 - e^{-0.84}) \] \[ P(2 \leq t \leq 4) = e^{-0.84} - e^{-1.68} \] By solving the above expression using a calculator, you would get the exact numerical likelihood value. **Note:** The example

the dice was rolled 3 times. find the probability that the results of the roll can be lengths of the sides of some triangle

At bedtime, you read three books to your kids, Walker, Michael, and Luke. Together, they have 32 books. If you decide to randomly choose three books, what is the probability that the three you choose will consist of each of their favorite books? Assume they have different favorites. Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.

Please give solution in correct and step by step detailed format thanku Don't give solution in image format A high school baseball player has a 0.151 batting average. In one game, he gets 9 at bats. What is the probability he will get at least 3 hits in the game?

if a baseball player's batting average is 0.320 find the probability that the player will get exactly 24 hits in 100 times at bat