The calculation of the expected value in slot machines is a theoretical calculation based on the probability of combinations, and produces statistics for each combination or win independently. We cannot determine the probability of wins occurring on multiple paylines at the same time.
I have found this sentence in a book but I did not understand it. Could you please explain me why we cannot determine the probability of wins occurring on multiple paylines at the same time?
For an individual slot machine, you can determine the odds of winning on multiple paylines, but it'll be a bit complicated and requires knowing the configuration of each individual reel. If all three reels are identical and have only one of each symbol, for example, hitting a 3-of-a-kind match on the center line means you also hit a 3-of-a-kind on the top and bottom lines. But in another configuration, the reels could be non-identical and set up so that hitting a 3-of-a-kind match on the center line makes it impossible to have a match on the top or bottom lines.
Without knowing specifics of the individual reel configurations, you can't determine the odds of hitting multiple paylines at once - some multiple payouts might be guaranteed, or impossible, or anywhere in between. In lieu of that information, one can approximate the value of a spin by just treating all payouts as independent, even though they generally are not. Most gambling machines show you the odds of hitting any individual payout, and from that alone, it is not possible to determine the joint probability of multiple payouts, so the approximation is all you can do.