# urgent help with math! probability question last one?

Suppose there are 10 friends in a study group. They are going to break up into two smaller groups of 5 people each, one of which will study at the café, and one of which will study at the library.

a.) If they are randomly divided into two groups of 5, then what is the probability that Alireza will be in the café group?

b.) Suppose Alireza and Beaumont don’t care whether they study at the library or at the café, but they do want to both be in the same group. What is the probability that this happens?

### 2 Answers

- atsuoLv 62 months ago
Let the group of 5 persons who study in the cafe be C, another group be L.

a.)

There are 10 persons and there are 5 persons in C. Alireza is 1 of 10 persons, so the probability that Alireza is in C is 5/10 = 1/2.

b.)

Let Alireza be in X (X is C or L). There are 9 persons except Alireza and there are 4 persons except Alireza in X, so the probability that Beaumont is in X is 4/9. Therefore, the probability that Alireza and Beaumont are in the same group is 4/9.

[Another method : combinatorial]

To divide 10 persons into C and L, we must choose 5 persons out of 10 persons. So there are 10C5 = 252 ways.

a.)

If Alireza is in C then other 4 persons out of 9 persons are chosen for C, so there are 9C4 = 126 ways. Therefore the answer is 126/252 = 1/2.

b.)

If Alireza and Beaumont are in X (X is C or L) then other 3 persons out of 8 persons are chosen for X, so there are 8C3 = 56 ways. And X is C or L, so there are 2 * 56 = 112 ways. Therefore the answer is 112/252 = 4/9.

- fcas80Lv 72 months ago
a. Either she is one group or the other, so 1/2.

b. If Ali is in group 1 with probability 1/2, the probability Beau is also in group 1 is 1/4, so this combination has probability 1/8. Similarly for group 2. 1/8 + 1/8 = 1/4.