Photo AI
In Galway, rain falls in the morning on $ rac{1}{3}$ of the school days in the year - Leaving Cert Mathematics - Question b - 2017
Question b
In Galway, rain falls in the morning on $ rac{1}{3}$ of the school days in the year. When it is raining the probability of heavy traffic is $ rac{1}{2}$. When it i... show full transcript
Worked Solution & Example Answer:In Galway, rain falls in the morning on $ rac{1}{3}$ of the school days in the year - Leaving Cert Mathematics - Question b - 2017
Step 1
Write the probability associated with each branch of the tree diagram and the probability of each outcome into the blank boxes provided.
Answer
To fill in the probabilities in the tree diagram, we can calculate each branch step-by-step:
-
Rain () Probability:
- P(R) = rac{1}{3}
- P(no R) = rac{2}{3}
-
Heavy Traffic () Probability when it Rains:
- P(T|R) = rac{1}{2}
- P(no T|R) = rac{1}{2}
-
Late () Probability when Rains with Heavy Traffic:
- P(L|R,T) = rac{1}{2}
- P(no L|R,T) = rac{1}{2}
-
Late () Probability when it Rains without Heavy Traffic:
- P(L|R,no T) = rac{5}{12}
- P(no L|R,no T) = rac{7}{12}
-
Heavy Traffic () Probability when it does not Rain:
- P(T|no R) = rac{1}{4}
- P(no T|no R) = rac{3}{4}
-
Late () Probability when it does not Rain with Heavy Traffic:
- P(L|no R,T) = rac{1}{12}
-
Late () Probability when it does not Rain without Heavy Traffic:
- P(L|no R,no T) = rac{1}{8}
-
Final Probabilities (Outcomes):
- P(R,T,L) = P(R) imes P(T|R) imes P(L|R,T) = rac{1}{3} imes rac{1}{2} imes rac{1}{2} = rac{1}{12}
- P(R,T,no L) = P(R) imes P(T|R) imes P(no L|R,T) = rac{1}{3} imes rac{1}{2} imes rac{1}{2} = rac{1}{12}
- P(R,no T,L) = P(R) imes P(no T|R) imes P(L|R,no T) = rac{1}{3} imes rac{1}{2} imes rac{5}{12} = rac{5}{72}
- P(R,no T,no L) = P(R) imes P(no T|R) imes P(no L|R,no T) = rac{1}{3} imes rac{1}{2} imes rac{7}{12} = rac{7}{72}
- P(no R,T,L) = P(no R) imes P(T|no R) imes P(L|no R,T) = rac{2}{3} imes rac{1}{4} imes rac{1}{12} = rac{1}{72}
- P(no R,T,no L) = P(no R) imes P(T|no R) imes P(no L|no R,T) = rac{2}{3} imes rac{1}{4} imes rac{11}{12} = rac{11}{72}
- P(no R,no T,L) = P(no R) imes P(no T|no R) imes P(L|no R,no T) = rac{2}{3} imes rac{3}{4} imes rac{1}{8} = rac{1}{32}
- P(no R,no T,no L) = P(no R) imes P(no T|no R) imes P(no L|no R,no T) = rac{2}{3} imes rac{3}{4} imes rac{7}{8} = rac{7}{32}
Step 2
On a random school day in Galway, find the probability of being late for school.
Answer
To find the total probability of being late for school, we sum up the probabilities from all relevant scenarios:
- For When it Rains:
- For When it does Not Rain:
Calculating these together, we find:
= rac{1}{12} + rac{5}{72} + rac{1}{72} + rac{1}{32}
Finding a common denominator (which is 144):
P(L) = rac{12}{144} + rac{10}{144} + rac{2}{144} + rac{4.5}{144} = rac{34.5}{144}
Simplifying this gives us:
= 0.239$\ ext{(approximately)}
Step 3
On a random school day in Galway, find the probability that it rained in the morning, given that you were late for school.
Answer
To find this probability, we can use Bayes' Theorem:
-
Finding :
-
This includes cases where it rained and you were late:
-
Calculating: P(R \cap L) = rac{1}{12} + rac{5}{72} = rac{6}{72} + rac{5}{72} = rac{11}{72}
-
-
Using the Total probability of being late from (ii):
- From previous answer, we have:
-
Calculating:
- This simplifies to:
- Further simplification leads to: