In one lottery game, contestants pick five numbers from 1 through 22 and have to match all five for the big prize (in any order). You'll get twice your money back if you match three out of five numbers. If you buy six tickets, what's the probability of matching three out of five numbers? If you buy six tickets, the probability of matching three out of five numbers is (Enter your answer as a fraction in lowest terms.)

Step 1 ：First, calculate the total number of possible combinations of 5 numbers from 22. This can be calculated using the combination formula \(C(n, r) = \frac{n!}{r!(n-r)!}\), where n is the total number of items, r is the number of items to choose, and ! denotes factorial. In this case, n=22 and r=5.

Step 2 ：Next, calculate the number of ways to choose 3 correct numbers and 2 incorrect numbers. The number of ways to choose 3 correct numbers from 5 is \(C(5, 3)\), and the number of ways to choose 2 incorrect numbers from the remaining 17 is \(C(17, 2)\).

Step 3 ：The probability of matching exactly 3 out of 5 numbers on a single ticket is then the number of ways to choose 3 correct and 2 incorrect numbers divided by the total number of combinations.

Step 4 ：Finally, since we are buying 6 tickets, we need to multiply this probability by 6 to get the final probability.

Step 5 ：Using these calculations, we find that the total number of combinations is 26334, the number of correct combinations is 1360, the probability of matching 3 out of 5 numbers on a single ticket is approximately 0.052, and the probability of matching 3 out of 5 numbers when buying six tickets is approximately 0.31.

Step 6 ：Final Answer: The probability of matching three out of five numbers when buying six tickets is \(\boxed{0.31}\).