Questions: Question 6 1 pts Which of the following could be used to find the probability that Tiger treats will sell less than 200 gallons of frozen yogurt on a summer day. Select all that apply. ∫ from 0 to 2 f(x) dx 1/2(2) · f(2) (area of the triangular region under f(x) from x=0 to x=2) 1-∫ from 2 to 3 f(x) dx

Question 6
1 pts

Which of the following could be used to find the probability that Tiger treats will sell less than 200 gallons of frozen yogurt on a summer day. Select all that apply.
∫ from 0 to 2 f(x) dx
1/2(2) · f(2) (area of the triangular region under f(x) from x=0 to x=2)
1-∫ from 2 to 3 f(x) dx

Transcript text: Question 6 1 pts Which of the following could be used to find the probability that Tiger treats will sell less than 200 gallons of frozen yogurt on a summer day. Select all that apply. $\int_{0}^{2} f(x) d x$ $\frac{1}{2}(2) \cdot f(2)$ (area of the triangular region under $f(x)$ from $x=0$ to $x=2$ ) $1-\int_{2}^{3} f(x) d x$

Solution

Solution Steps

To find the probability that Tiger treats will sell less than 200 gallons of frozen yogurt, we need to consider the integral of the probability density function $ f(x) $ from 0 to 2, as this represents the cumulative probability up to 200 gallons. The other options involve calculations that do not directly represent the cumulative probability up to 200 gallons.

Step 1: Understanding the Problem

The problem asks us to determine which of the given expressions could be used to find the probability that Tiger treats will sell less than 200 gallons of frozen yogurt on a summer day. The expressions involve integrals and an area calculation, which are typically used to find probabilities in continuous probability distributions.

Step 2: Analyzing the Options

Let's analyze each option to see if it can be used to find the probability of selling less than 200 gallons.

Option 1: $\int_{0}^{2} f(x) \, dx$

This integral represents the area under the probability density function $f(x)$ from $x = 0$ to $x = 2$. If $f(x)$ is the probability density function for the number of gallons sold, then this integral gives the probability of selling between 0 and 2 units (assuming the units are in hundreds of gallons, i.e., 200 gallons).
Option 2: $\frac{1}{2}(2) \cdot f(2)$

This expression calculates the area of a triangular region under $f(x)$ from $x = 0$ to $x = 2$. This is not a standard method for calculating probabilities unless the function $f(x)$ is specifically defined as a triangular distribution over this interval. Without additional context, this option is not generally applicable for finding the probability of selling less than 200 gallons.
Option 3: $1-\int_{2}^{3} f(x) \, dx$

This expression calculates the probability of selling more than 2 units (or 200 gallons) and subtracts it from 1, which gives the probability of selling less than 2 units. This is a valid method for finding the probability of selling less than 200 gallons.