In a lottery game, a machine has 48 balls marked with numbers from 1-48 and six of them are taken out at random. A player picks six of these numbers, trying to match the six balls that come out of the machine in any order. If exactly 4 of those 6 numbers match those drawn, the player wins third prize. What is the probability of winning this prize?
How would I solve this problem?
Best Answer
There are $\ {6\choose 4}\ $ subsets of the player's $6$ cards which could be the one exactly matched, and the matching set could occur in any of $\ {6\choose 4}\ $ sets of positions of the $6$-ball draw. There are thus $\ {6\choose 4}^2\ $ different ways in which the match could occur, each with the same probability of $$ \frac{4\cdot3\cdot2\cdot1\cdot42\cdot41}{48\cdot47\cdot46\cdot45\cdot44\cdot43}\ . $$ Thus, the probability of such an exact match occurring is $$ {6\choose 4}^2\frac{4\cdot3\cdot2\cdot1\cdot42\cdot41}{48\cdot47\cdot46\cdot45\cdot44\cdot43}=\frac{4305}{4090504}\approx0.00105 $$