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February 26th, 2015, 08:43 AM   #1
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Find the probability the student is late

A student goes to school by bus, taxi or motorcycle. The probability that he travels by motorcycle is 0.44 and he is equally likely to take a bus or a taxi. The probability that he is late for school if he goes by bus, taxi or motorcycle is $\displaystyle \frac{1}{5}$, $\displaystyle \frac{1}{6}$ or$\displaystyle \frac{1}{10}$ respectively. Calculate the probability that

a)he is late for school on a randomly chosen school day
B)he hoes to school by bus if he is late for school
c)he is not late for school if he goes to school by bus or motorcycle.
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February 26th, 2015, 10:55 AM   #2
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$$\newcommand{\pr}[1]{\Pr{\left(#1\right)}}
\begin{aligned}
\pr{ \text{late} } &= \pr{ \text{travels by motorcyle} }\pr{ \text{motorcyle is late} }+ \pr{ \text{travels by bus} }\pr{ \text{bus is late} }+ \pr{ \text{travels by taxi} }\pr{ \text{taxi is late} } \\
\pr{ \text{goes by bus} | \text{late} } &= {\pr{ \text{goes by bus and is late} } \over \pr{\text{late}}} \\
\pr{ \text{goes by bus} | \text{late} } &= {\pr{ \text{goes by bus and is late} } \over \pr{\text{late}}} \\
\pr{ \text{not late|goes by bus or taxi} } &= { \pr{ \text{not late and goes by bus or taxi} } \over \pr{ \text{goes by bus or taxi} } }
= { \pr{ \text{not late and goes by bus} } + \pr{ \text{not late and goes by taxi} } \over \pr{ \text{goes by bus} } + \pr{\text{goes by taxi} } } \\
&= { \pr{\text{goes by bus}}\left(1-\pr{\text{bus is late}}\right) + \pr{\text{goes by taxi}}\left(1-\pr{\text{taxi is late}}\right) \over
\pr{ \text{goes by bus} } + \pr{\text{goes by taxi} } }
\end{aligned} $$
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