You are asked to determine if the two events "a randomly selected student has asthma" and "a randomly selected student has a household member who smokes" are independent. what probabilities could you calculate to answer this question?

Asthma is more likely to occur in someone who lives in a household with at least one person who smokes

Let's define the two events A and B to be...
event A = student has asthma
event B = at least one household member smokes

To find the probability of event A happening, we count the total number of people who have asthma out of the total in the survey. So we simply look at the "student has asthma" row and look at the "total" column. This value is 182. There are 182 people who have asthma. This is out of 937 (bottom right corner).

Divide the values: 182/937 = 0.194237
This value is approximate

So P(A) = 0.194237 approximately
The P means probability
P(A) = probability of event A happening



Now compute P(B)
P(B) = 395/937
P(B) = 0.421558
Notice I'm dividing the total value in the "at least one member smokes" column over the grand total of 937



Next we compute P(A and B) which is the probability both events occur at the same time. 

P(A and B) = 113/937
P(A and B) = 0.120598
The 113 is the amount of people who have asthma AND have at least one household member that smokes (row1, column2)



If events A and B were independent, then P(A)*P(B) should be equal to P(A and B)

Let's see if that is the case
P(A and B) = P(A)*P(B)
0.120598 = P(A)*P(B)
0.120598 = 0.194237*0.421558
0.120598 = 0.08188216
we get two different results so P(A and B) = P(A)*P(B) is not true

Therefore, A and B are not independent events. They are said to be dependent events.

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