A machine randomly generates one of the nine numbers 1, 2, 3, … , 9 with equal likelihood. What is the probability that when Tsuni uses this machine to generate four numbers their product is divisible by 4? Express your answer as a common fraction.
+2666 Note that for the product to be divisible by 4, there need to be at least 2 even numbers.
The probability of rolling only odd numbers is \({5 \over 9}^4 = {1025 \over 6561}\)
The probability of rolling exactly 1 even number is \(4 \times 5^3 \times {4 \choose 3} = 2000\)
So, the probability is \(1 - ({1025 \over 6561} + {2000 \over 6561}) = \text{_____}\)
BuilderBoi: A computer code gives the following numbers:
9^4 ==6,561 permutations. [This is based on the assumption that numbers are replaced at each draw]
The computer calculates:
The probability is; 4,936 / 6,561
+2666 REVISED ANSWER: There are \(9^4 = 6561\) ways to get the numbers.
The easiest way to solve this is to use complementary counting, in which we count what we don't want, and subtract that from 1
Note that the probability of getting an odd number is \({5 \over 9}^4 = {625 \over 6561}\)
The probability of getting a 2 or 6 is \({2 \times 5^3 \times {4 \choose 3} \over 6561} = {1000 \over 6561}\) note that 4 and 8 make the product divisible by 4.
So, the total probability is \(1 - ({625 \over 6561} + {1000 \over 6561} ) = \color{brown}\boxed{4936 \over 6561}\)
