Let's solve part B: $P(\text{not}(A \text{ and } C))$. Here's the step-by-step process:

Step 1: Understand the Problem

• $A$: early out
• $B$: late start
• $C$: boys
• $D$: girls
• $P(A \text{ and } C)$: the probability of being in the early out category and a boy.

"not(A and C)" means we want the probability of everything except being a boy and in the early out group.

Step 2: Identify $A \text{ and } C$

From the table:

• $A \text{ (early out)}$ and $C \text{ (boys)}$: 14 students.

Thus, the number of outcomes for $A \text{ and } C$ is $14$.

Step 3: Calculate $P(A \text{ and } C)$

The total number of students is $40$. Therefore:
$P(A \text{ and } C) = \frac{14}{40}.$

Simplify the fraction:
$P(A \text{ and } C) = \frac{7}{20}.$

Step 4: Calculate $P(\text{not}(A \text{ and } C))$

The complement of $A \text{ and } C$ is:
$P(\text{not}(A \text{ and } C)) = 1 - P(A \text{ and } C).$

Substitute $P(A \text{ and } C) = \frac{7}{20}$:
$P(\text{not}(A \text{ and } C)) = 1 - \frac{7}{20}.$

Convert $1$ to $\frac{20}{20}$:
$P(\text{not}(A \text{ and } C)) = \frac{20}{20} - \frac{7}{20} = \frac{13}{20}.$

Step 5: Convert to Decimal

To convert $\frac{13}{20}$ to a decimal:
$\frac{13}{20} = 0.65.$

Final Answer:

• Simplified Fraction: $\frac{13}{20}$
• Decimal: $0.65$

Let me know if you need clarification or further explanation!

Related Questions:

1. What is $P(\text{not}(B \text{ and } D))$?
2. Calculate $P(A \text{ or } D)$.
3. Find the probability of being a girl and late start.
4. What is $P(A \text{ and } D)$?
5. Calculate the complement of $P(B \text{ and } C)$.

Tip: Always remember that probabilities of complements add up to $1$. Use this when calculating "not" events!