in this problem, round to four decimal places. hint: see this problem +. the probability of a dry day is…

in this problem, round to four decimal places. hint: see this problem +. 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

in this problem, round to four decimal places. hint: see this problem +. 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)

$P(\text{wet}) = 1 - P(\text{dry}) = 1 - \frac{2}{9} = \frac{7}{9}$

Step2: Apply total probability for train

$P(\text{train}) = P(\text{train|dry})P(\text{dry}) + P(\text{train|wet})P(\text{wet})$

Step3: Substitute values

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

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 and simplify

$\frac{2}{27} + \frac{77}{171} = \frac{38}{513} + \frac{231}{513} = \frac{269}{513} \approx 0.5244$

Answer:

0.5244