Conditional probability and independence

James is interested in weather conditions and weather the downtown train he sometimes takes runs on time for a year James records weather each day is sunny cloudy rainy or snowy as well as weather this train arrives on time or is delayed his results are displayed in the table below alright this is interesting these columns on time delayed and the total so for example when it was sunny there's a total of 170 sunny days that year 167 of which the train was on time three of which the train was delayed and we could look at that by the different types of weather conditions and then they say for these days are the events delayed and snowy independent so to think about this and remember we're only going to be able to figure out experimental probabilities and you should always view experimental probabilities it somewhat suspect the more experiments you're able to take the more likely it is to approximate the true theoretical probability but there's always some chance that they might be different or even quite different let's use this data to try to calculate the experimental probability so the key question here is what is the probability that the train is delayed and then we want to think about what is the probability that the train is delayed given that it is snowy if we knew the theoretical probabilities and if they were exactly the same if the probability of being delayed was exactly the same as the probability of being delayed given snowy then being delayed or being snowy would be independent but if we knew the theoretical probabilities and the probability of being delayed given snowy were different than the probability of being delayed then we would not say that these are independent variables now we don't know the theoretical probabilities we're just going to calculate the experimental probabilities and we do have a good number of experiments here so if these are quite different I would feel confident saying that they are dependent if they are pretty close with the experimental probability I would say that it would be hard to make the statement that they are dependent and that you would probably lean towards independence but let's calculate this what is the probability that the train is just delayed pause this video and try to figure that out well let's see if we just think in general we have a total of 365 trials or 365 experiments and of them the train was delayed 35 times now what's the probability that the Train is delayed given that it is Snowie pause the video and try to figure that out well let's see we have a total of 20 snowy days and we are delayed 12 of those 20 snowy days and so this is going to be a probability 12 20th is the same thing as if we multiply both the numerator and the denominator by 5 this is a 60% probability or I could say a 0.6 probability of being delayed when it is snowy this is of course in experimental probability which is much higher than this this is less than 10% right over here this right over here is less than 0.1 I could get a calculator to calculate it exactly it'll be 9 point something percent or 0.9 something but clearly this you are much more likely at least from the experimental data it seems like you know a much higher proportion of your snowy days are delayed than just general days in general than just general days and so based on this data because the experimental probability of being delayed given snowy is so much higher than the experimental probability of just being delayed I would make the statement that these are not independent so for these days are the events delayed and snowy independent no