+400 A machine randomly generates one of the nine numbers 1, 2, 3, \dots, 9 with equal likelihood. What is the probability that when Tsuni uses this machine to generate four numbers their product is divisible by 2?
Express your answer as a common fraction.
+1632 We can do complimentary counting, where you have the total choices (1) minus the wrong choices.
The only way that the machine can generate four numbers where the product is NOT divisible by 2 is if and only if they are all odd numbers. From 1 through 9, there are 5 odd and 4 even numbers.
For the first number generated, there is a 5/9 chance it is odd, and for the next number 5/9, the third 5/9, and the last 5/9 chance again.
Thus, the probability is \({5^4\over{9^4}}={625\over{6561}}\) for generating four numbers with a product not divisible by 2.
To calculate the probability that the machine will generate four numbers where their product is divisible by 2, you just subtract 625/6561 from 1, which equals to = \(5936/6561\) probability.
