Probability of two friends being in the same group

If 10 people are divided into 2 equal groups(each group has 5 people) whats the probability that two friends A and B are in the same group?

Solution 1:

Hint: A is definitely in A's group. How many others are in A's group?

Solution 2:

Method 1

You can count all the possible groups:

$$\binom{10}{5} = 252 \textrm{ groups}$$

Pick 3 people to be in a group with A and B: $$\binom{8}{3} = 56 \textrm{ groups with A and B together}$$

Alternatively, we might select a 5-person group with neither A nor B: $$\binom{8}{5} = 56 \textrm{ groups with A and B together}$$

This gives us a total of $112$ arrangements with A and B together:

$$\frac{112}{252} = \frac{4}{9}$$

Method 2

Fix A in any group. There are 4 spots left in that group, and 5 in the other group, so there must be a $\frac{4}{9}$ chance of B being placed in the group with A