Question

the following two - way table shows the distribution of daily traffic and weather issues in a certain large city.

weather conditions	daily traffic heavy traffic	daily traffic light traffic	total
good weather	55	15	75
total	80	20	100

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?
yes, p(a)=p(a|b)
yes, p(a)=p(b|a)
no, p(a)≠p(a|b)
no, p(a)=p(b|a)

Explanation:

Step1: Calculate P(A)

$P(A)=\frac{\text{Number of heavy - traffic cases}}{\text{Total cases}}=\frac{80}{100} = 0.8$

Step2: Calculate P(A|B)

$P(A|B)=\frac{P(A\cap B)}{P(B)}$, where $P(A\cap B)=\frac{25}{100}$ and $P(B)=\frac{30}{100}$. So $P(A|B)=\frac{\frac{25}{100}}{\frac{30}{100}}=\frac{25}{30}=\frac{5}{6}\approx0.833$

Step3: Check independence

Two events A and B are independent if $P(A) = P(A|B)$. Since $0.8
eq\frac{5}{6}\approx0.833$, $P(A)
eq P(A|B)$

Answer:

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