2) The masses, in grams, of large bags of sugar and small bags of sugar are denoted by $$ X $$ and $$ Y $$ respectively, where $$ X \sim N\left(5.1,0.2^{2}\right) $$ and $$ Y \sim N\left(2.5,0.12^{2}\right) $$. Find the probability that the mass of a randomly chosen large bag is less than twice the mass of a randomly chosen small bag. [5] 3) The masses, in kilograms, of cartons of sugar and cartons of flour have the distributions N(78.8, $$ \left.12.6^{2}\right) $$ and $$ N\left(62.0,10.0^{2}\right) $$ respectively. a) The standard load for a certain crane is 8 cartons of sugar and 3 cartons of flour. The maximum load that can be carried safely by the crane is 900 kg. Stating a necessary assumption, find the percentage of standard loads that will exceed the maximum safe load. [5]

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### Problem 2

**Given:**
- Mass of a large bag of sugar, $$ X \sim N(5.1\, \text{g},\, 0.2^2\, \text{g}^2) $$
- Mass of a small bag of sugar, $$ Y \sim N(2.5\, \text{g},\, 0.12^2\, \text{g}^2) $$

**Objective:**
Find the probability that the mass of a randomly chosen large bag is less than twice the mass of a randomly chosen small bag, i.e., $$ P(X < 2Y) $$.

**Solution:**

1. **Define the Event:**
   $$
   P(X < 2Y) = P(X - 2Y < 0)
   $$
   
2. **Determine the Distribution of $$ Z = X - 2Y $$:**
   Since $$ X $$ and $$ Y $$ are independent normal random variables, $$ Z $$ will also follow a normal distribution.

- **Mean of $$ Z $$:**
     $$
     \mu_Z = E[X] - 2E[Y] = 5.1 - 2 \times 2.5 = 5.1 - 5.0 = 0.1\, \text{g}
     $$
     
   - **Variance of $$ Z $$:**
     $$
     \sigma_Z^2 = \text{Var}(X) + 4\,\text{Var}(Y) = 0.2^2 + 4 \times 0.12^2 = 0.04 + 4 \times 0.0144 = 0.04 + 0.0576 = 0.0976\, \text{g}^2
     $$
     
   - **Standard Deviation of $$ Z $$:**
     $$
     \sigma_Z = \sqrt{0.0976} \approx 0.312\, \text{g}
     $$
     
   Thus, $$ Z = X - 2Y \sim N(0.1\, \text{g},\, 0.312^2\, \text{g}^2) $$.

3. **Standardize the Variable to Find the Probability:**
   We need to find $$ P(Z < 0) $$.

- **Calculate the Z-score:**
     $$
     Z_{\text{score}} = \frac{0 - \mu_Z}{\sigma_Z} = \frac{0 - 0.1}{0.312} \approx -0.3205
     $$
     
   - **Find the Probability Using Standard Normal Distribution:**
     $$
     P(Z < 0) = P\left(Z < -0.3205\right) = \Phi(-0.3205)
     $$
     Where $$ \Phi $$ is the cumulative distribution function (CDF) of the standard normal distribution.

Using standard normal distribution tables or a calculator:
     $$
     \Phi(-0.3205) \approx 0.3745 \text{ or } 37.45\%
     $$

**Conclusion:**
The probability that the mass of a randomly chosen large bag of sugar is less than twice the mass of a randomly chosen small bag is approximately **37.5%**.
The probability that a large bag of sugar weighs less than twice a small bag is about 37.5%. [Example 5] question: Solve the equation $$ 3x - 2 = 7 $$ for x. full answer: Solve the equation $$ 3x - 2 = 7 $$ for x, we get $$ x = 3 $$. So, the solution of the equation $$ 3x - 2 = 7 $$ is $$ x = 3 $$. simplify answer: $$ x = 3 $$.

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