Question

the following two - way table shows the distribution of daily traffic and weather issues in a certain large city. suppose a day from this city is selected at random. let event ( a=) heavy traffic and event ( b=) bad weather. are events ( a ) and ( b ) independent?
no, ( p(a)=p(b|a) ).
no, ( p(a)
eq p(a|b) ).
yes, ( p(a)=p(a|b) ).
yes, ( p(a)
eq p(b|a) ).

Explanation:

Step1: Calculate \(P(A)\)

The formula for probability is \(P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
For event \(A\) (heavy traffic), the number of favorable outcomes is \(80\) and the total number of outcomes is \(100\).
So, \(P(A)=\frac{80}{100} = 0.8\)

Step2: Calculate \(P(A|B)\)

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\).
\(P(A\cap B)=\frac{25}{100}\) (number of days with both heavy traffic and bad weather divided by total days) and \(P(B)=\frac{30}{100}\) (number of days with bad weather divided by total days).
Then \(P(A|B)=\frac{\frac{25}{100}}{\frac{30}{100}}=\frac{25}{30}\approx0.833\)

Step3: Check independence condition

Two events \(A\) and \(B\) are independent if \(P(A) = P(A|B)\)
Since \(P(A)=0.8\) and \(P(A|B)\approx0.833\), \(P(A)
eq P(A|B)\)

Answer:

No, \(P(A)
eq P(A|B)\)