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
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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
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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)
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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.
