Probability

We want to find the chance that the selected girl comes from F2.
Step 1: Find the chance of picking each family.
There are three families, so the chance of picking any one family ( F1,F2, or F3 ) is ‌

1

3

.
Step 2: Find the chance of picking a girl from each family.
For F1 (2 boys, 1 girl): Probability of picking a girl =‌

1

3

For F2 (1 boy, 2 girls): Probability of picking a girl =‌

2

3

For F3 (1 boy, 1 girl): Probability of picking a girl =‌

1

2

Step 3: Find the total probability of picking a girl (from any family).
To get the total chance of picking a girl, add up the chance for each family:
‌P(G)=P(G∣F1)⋅P(F1)+P(G∣F2)⋅P(F2)+P(G∣F3)⋅P(F3)
‌=‌

1

3

⋅‌

1

3

+‌

2

3

⋅‌

1

3

+‌

1

2

⋅‌

1

3

‌=‌

1

9

+‌

2

9

+‌

1

6

=‌

1+2+3

18

=‌

6

12

=‌

1

2

Step 4: Use Bayes' Theorem to find the chance that the girl is from F2.
Bayes' Theorem tells us: P(F2∣G)=‌

P(G∣F2)⋅P(F2)

P(G)

=‌

‌ 2 3 ⋅ 1 3

1 2

=‌

‌ 2 9

1 2

=‌

2

9

×2=‌

4

9

Final Answer:
The probability that the selected girl is from F2 is ‌

4

9

.