Question

james is interested in the relationship between weather conditions and whether the downtown train he sometimes takes runs on time. for a year, james records whether each day is sunny, cloudy, rainy, or snowy, as well as whether this train arrives on time or is delayed. his results are displayed in the table below.
weather condition on time delayed total
sunny 167 3 170
cloudy 115 5 120
rainy 40 15 55
snowy 8 12 20
total 330 35 365
give the conditional distribution of weather condition for delayed trains. round your answers to the nearest tenth of a percent.

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. In the context of a two - way table for finding the conditional distribution of weather condition for delayed trains, if $A$ is the event of a particular weather condition and $B$ is the event of the train being delayed, then the proportion for a particular weather condition in the conditional distribution of weather for delayed trains is $\frac{\text{Number of delayed trains in that weather condition}}{\text{Total number of delayed trains}}$.

Step2: Calculate for sunny weather

The number of delayed trains in sunny weather is 3, and the total number of delayed trains is 35. So the proportion is $\frac{3}{35}\approx 0.086$, as a percentage it is $0.086\times100 = 8.6\%$.

Step3: Calculate for cloudy weather

The number of delayed trains in cloudy weather is 5. So the proportion is $\frac{5}{35}\approx0.143$, as a percentage it is $0.143\times 100 = 14.3\%$.

Step4: Calculate for rainy weather

The number of delayed trains in rainy weather is 15. So the proportion is $\frac{15}{35}\approx0.429$, as a percentage it is $0.429\times 100 = 42.9\%$.

Step5: Calculate for snowy weather

The number of delayed trains in snowy weather is 12. So the proportion is $\frac{12}{35}\approx0.343$, as a percentage it is $0.343\times 100 = 34.3\%$.

Answer:

Sunny: 8.6%, Cloudy: 14.3%, Rainy: 42.9%, Snowy: 34.3%