With a perfectly balanced roulette wheel, in the long run, red numbers should turn up 18 times in 38. to test its wheel, one casino records the results of 3,800 plays, finding 1890 red numbers. is that too many reds? or chance variation?

The red numbers will turn up 18 times out of 38.
So the theoretical probability of getting a red = 18/38 = 9/19

The experiment results in 1890 reds out of 3800.
So experimental probability of getting the red = 1890/3800 = 189/380

We can use z-score to see if the number of reds is unusually large or not.
z score will be:

[tex] \frac{ \frac{189}{380} - \frac{9}{19} }{ \sqrt{ \frac{ \frac{9}{19} * \frac{10}{19} }{3800} } }=2.763 [/tex]

The probability from the z table for this z score comes out to be 99.71% 

The probability below 5% and above 95% is said to be unsual. This mean, the event occurred by chance and is not within the usual range of values Based on this we can say, yes there were too many reds in the experiment done by Casino.