\#27 i ROBLEM SOLVING There are 8 females and 10 males in a class. a. What is the theoretical probability that a randomly chosen student is female? The theoretical probability is One week later, there are 27 students in the class. The theoretical probability that a andomly chosen student is a female is the same as last week. How many males joined the lass?

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UpStudy AI · Accepted Answer

**Step 1. Determine the original probability that a randomly chosen student is female.**

There are 8 females and 10 males, so the total number of students is  
$$
8 + 10 = 18.
$$

Thus, the probability that a randomly chosen student is female is  
$$
\frac{8}{18} = \frac{4}{9}.
$$

---

**Step 2. Analyze the situation one week later.**

After one week, there are 27 students in the class. Let the number of new females be $$ f $$ and the number of new males be $$ m $$. The new totals are:
- Total females = $$8 + f$$.
- Total males = $$10 + m$$.

The sum of the new students is
$$
f + m = 27 - 18 = 9.
$$

We are given that the theoretical probability that a randomly chosen student is female remains the same, i.e.,
$$
\frac{8 + f}{27} = \frac{4}{9}.
$$

---

**Step 3. Solve for the number of new females $$ f $$.**

Multiply both sides of the equation by 27:
$$
8 + f = \frac{4}{9} \times 27.
$$

Simplify the right-hand side:
$$
\frac{4}{9} \times 27 = 4 \times 3 = 12.
$$

Thus,
$$
8 + f = 12 \quad \Longrightarrow \quad f = 12 - 8 = 4.
$$

---

**Step 4. Determine the number of new males $$ m $$.**

Since the total number of new students is 9,
$$
m = 9 - f = 9 - 4 = 5.
$$

---

**Final Answers:**

- The theoretical probability that a randomly chosen student is female is  
  $$
  \frac{4}{9}.
  $$
- The number of males who joined the class is  
  $$
  5.
  $$
The probability that a randomly chosen student is female is 4/9. Five males joined the class.

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