Question

My best friend is throwing a birthday party for me. She is going to invite 5 of my friends, chosen randomly. (She does not matter in this question)

I’m not very popular, so I only have 10 friends, and 3 of them are my family members.

What my best friend doesn’t know is that my family hate each other. If 2 or more of my family show up at the party, a fight is going to happen.

What is the probability that a fight breaks out at my party?

Answer 1

Answer 2

Solution:

Given:

• Total friends = 10

• Family members = 3

• Non-family friends = 7

• Invites = 5 (randomly chosen)

Fight condition: 2+ family members invited.

Probability Calculation:

1. Total ways to choose 5 friends:

   $$\left(\genfrac{}{}{0px}{}{10}{5}\right)=252$$

2. Favorable outcomes (fight scenarios):

• Exactly 2 family + 3 non-family:

  $$\left(\genfrac{}{}{0px}{}{3}{2}\right)\times \left(\genfrac{}{}{0px}{}{7}{3}\right)=3\times 35=105$$

• Exactly 3 family + 2 non-family:

  $$\left(\genfrac{}{}{0px}{}{3}{3}\right)\times \left(\genfrac{}{}{0px}{}{7}{2}\right)=1\times 21=21$$

• Total favorable: $105+21=126$

4. Probability of fight:

   $$\frac{126}{252}=0.5(\text{50\%})$$

Answer:
The probability a fight breaks out is 50%.