A company employs 5 males and 5 females in their technology team. If four people are selected
for a new training program, and 𝑃𝑃� represents the proportion of females chosen, then
Pr�𝑃𝑃� > 0.5� is:


The problem asks to calculate the probability \( \text{Pr}(P_f > 0.5) \), where \( P_f \) represents the proportion of females chosen in a selection of four people from a group of 5 males and 5 females.

### Step 1: Total number of ways to select 4 people
The total number of ways to select 4 people from 10 (5 males and 5 females) is given by the combination formula:
\[
\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
\]

### Step 2: Scenarios where \( P_f > 0.5 \)
For \( P_f > 0.5 \), we need more than half of the selected group to be females. In a group of 4 people, this means we must select at least 3 females. Thus, we need to consider two cases:
- 3 females and 1 male
- 4 females and 0 males

#### Case 1: Selecting 3 females and 1 male
The number of ways to select 3 females from 5 is:
\[
\binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10
\]
The number of ways to select 1 male from 5 is:
\[
\binom{5}{1} = 5
\]
Thus, the total number of ways to select 3 females and 1 male is:
\[
\binom{5}{3} \times \binom{5}{1} = 10 \times 5 = 50
\]

#### Case 2: Selecting 4 females and 0 males
The number of ways to select 4 females from 5 is:
\[
\binom{5}{4} = 5
\]
The number of ways to select 0 males from 5 is:
\[
\binom{5}{0} = 1
\]
Thus, the total number of ways to select 4 females and 0 males is:
\[
\binom{5}{4} \times \binom{5}{0} = 5 \times 1 = 5
\]

### Step 3: Total number of favorable outcomes
The total number of favorable outcomes where \( P_f > 0.5 \) is the sum of the favorable outcomes from both cases:
\[
50 \text{ (from case 1)} + 5 \text{ (from case 2)} = 55
\]

### Step 4: Probability calculation
The probability is the ratio of favorable outcomes to the total number of possible outcomes:
\[
\text{Pr}(P_f > 0.5) = \frac{55}{210} \approx 0.2619
\]

Thus, the probability that the proportion of females chosen is greater than 0.5 is approximately **0.2619**.

Would you like more details on any step? Here are some related questions:

1. How does the combination formula work?
2. What is the significance of \( P_f > 0.5 \) in this context?
3. How can we compute combinations more efficiently?
4. How does probability change if the group size increases?
5. How would the probability change if the ratio of males to females was different?

**Tip:** When calculating combinations, it's helpful to break the problem into smaller cases, especially when dealing with multiple selection scenarios.

A company employs 5 males and 5 females in their technology team. If four people are selected for a new training program, and P_f represents the proportion of females chosen, then Pr(P_f > 0.5) is:

Calculate the probability that more than 50% of selected team members are females from a group of 5 males and 5 females. Learn to use combinatorics and probability theory in this detailed step-by-step guide. Suitable for Grades 10-12.

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