# Probability of 2 students being chosen the both have under 100 books at home

Suppose we select two students at random from the class of fifteen. What is the probability that both students chosen have less then 100 books at home?

Data provided is the amount of books each student has at home. There are 6 out of the 15 students that have less than 100 books:

33, 47, 48, 52, 56, 66, 173, 186, 251, 300, 323, 417, 450, 1280, 2000

I tried doing 6/15 x 5/14 = 30/210 or a 14% chance. Is this correct?

Thank you!

• @MarcusM with "order matters", the order of what are you referring to? – Slug Pue Jul 31 '15 at 2:53
• @MarcusM Sorry that's just how the question was worded but I am sure of it that the order doesn't matter – coopwatts Jul 31 '15 at 2:55
• @MarcusM The answer does not imply that order matters. If you compute it with binomial coefficients instead you'll get the same answer. – Matt Samuel Jul 31 '15 at 2:57
• Oops! My bad. The $2$'s cancel. – Marcus M Jul 31 '15 at 3:08

$L_1$ is the event that the first student has less than $100$ books,
$L_2$ is the event that the second has less than $100$ books.
$$P(L_1\cap L_2) = P(L_1)\cdot P(L_2\ |\ L_1)$$ $$= \frac{6}{15}\cdot \frac{5}{14}$$