Solved – Probability to get an even number while rolling a 4 dice


$4$ dice are rolled about 200 times and it is noted how many times an even number showed up.( 0 to 4) . I tried to solve like this but I don't understand further where I made a mistake , probability that I get 0 times even number is equal to $(0.5)^4 *200$ it means that all four times I got an odd number because while rolling a die one time I have chance $\frac{3}{6} = 0.5$ to get an even number and also the same for odd numbers but when I solve that i get $1$ time an even number which I think is equal to getting 3 times an odd, so I do it like this $(0.5)^3*200$ and I get 25 while the correct result is $50$ . Can someone check where I made a mistake? I need this for running a Chi-squared test as I am trying to calculate the expected frequency.
Extra Info:
It is not self study nor assignment, as exam preparation I searched online for difficult examples. I am trying to calculate the expected frequency , for $0,1,2,3,4$ are number of times I get an even number. When I roll 1 die , I get the probability of getting an odd number $\frac{3}{6}$ because all 4 dice are independent and if all 4 dice have odd numbers then it is equal to getting 0 even numbers so because of independence I multiplied them with each other , $(1/6*1/6*1/6*1/6)$. In total there are no 200 dice but 4 dice are rolled for 200 times , so I mutiplied this by 200 .

Best Answer

probability that i get 0 times even number is equal to (0.5)^4∗200

No, that would be the expected count of the same sequence of outcomes, after 200 throws. But there are more than 1 sequence, leading to an even outcome. This is where it goes wrong.

To obtain your frequencies, you need to evaulate first how many different sequences you can throw at all, irrespective of odd or even outcomes.

Then look for the sequences which give odd outcomes, which you already found: sequences with exactly 1 odd or sequences with exactly 3 odd (or 1 even).

The probability of finding 1 odd outcome is the number of sequences to throw 1 odd:

oeee, eoee, eeoe, eeeo

divided by the total number of sequences.

The total number of sequences is $2^4=16$.

Since the are 4 sequences with 1 odd, the probability of finding 1 odd is $4/16=1/4$. So the expected frequency will be $200\cdot\,1/4=50$.