the probability of a dry day is $\frac{2}{9}$. if it is a dry day, the probability that opal cycles to work…

the probability of a dry day is $\frac{2}{9}$. if it is a dry day, the probability that opal cycles to work is $\frac{5}{12}$, that she drives to work is $\frac{1}{4}$, and that she takes the train to work is $\frac{1}{3}$. if it is a wet day, the probability that opal cycles to work is $\frac{4}{19}$, that she drives to work is $\frac{4}{19}$, and that she takes the train to work is $\frac{11}{19}$. determine the probability that opal takes the train to work. (create a tree diagram to help you) you may enter a calculation that leads to your answer

the probability of a dry day is $\frac{2}{9}$. if it is a dry day, the probability that opal cycles to work is $\frac{5}{12}$, that she drives to work is $\frac{1}{4}$, and that she takes the train to work is $\frac{1}{3}$. if it is a wet day, the probability that opal cycles to work is $\frac{4}{19}$, that she drives to work is $\frac{4}{19}$, and that she takes the train to work is $\frac{11}{19}$. determine the probability that opal takes the train to work. (create a tree diagram to help you) you may enter a calculation that leads to your answer

Answer

Explanation:

Step1: Calculate P(wet day)

( P(W) = 1 - P(D) = 1 - \frac{2}{9} = \frac{7}{9} )

Step2: Apply total probability for train

( P(T) = P(T|D)P(D) + P(T|W)P(W) )

Step3: Substitute values

( P(T) = \left( \frac{1}{3} \times \frac{2}{9} \right) + \left( \frac{11}{19} \times \frac{7}{9} \right) )

Step4: Compute each term

( \frac{1}{3} \times \frac{2}{9} = \frac{2}{27} ), ( \frac{11}{19} \times \frac{7}{9} = \frac{77}{171} )

Step5: Add terms (common denominator 513)

( \frac{2}{27} = \frac{38}{513} ), ( \frac{77}{171} = \frac{231}{513} ), ( P(T) = \frac{38 + 231}{513} = \frac{269}{513} )

Answer:

(\frac{269}{513})