Question

ion 3. (20 pts.) William (the student) remembers to take his umbrella with him 80% of th days. It rains on 30% of the days when he remembers to take his untren and it rains on 60% of the days when he forgets to take his umbrela wh is the probability that he remembers his umbrella when t rains?

Answer 1

Solution : ( 3 )

$\mathrm{Let\:A=William\:remembers\:to \:take\:his\:umbrella}$

$\mathrm{and}$

$\mathrm{B=Its\:rain}$

$\mathrm{Using\:Bayes's\:theorem\::}$

$\mathrm{P(A\mid B)=\frac{P(A)P(B\mid A)}{P(B)}}$

$\mathrm{Here,}$

$\mathrm{P(A)=80\%=\frac{80}{100}=0.8}$

$\mathrm{P(B)=30\%*80\%+60\%*20\%=\frac{30}{100}*\frac{80}{100}+\frac{60}{100}*\frac{20}{100}=0.3*0.8+0.6*0.2=0.24+0.12=0.36}$$\mathrm{P(A\mid B)=30\%=\frac{30}{100}=0.3}$

$\mathrm{Now,}$

$\mathrm{Therefore,\quad\:P(A\mid B)=\frac{P(A)P(B\mid A)}{P(B)}=\frac{0.8*0.3}{0.36}=\frac{0.24}{0.36}=\frac{2}{3}}$

$\mathrm{Therefore,\quad\:the\:probability\:that\:he\:remembers\:his\:umbrella\:when\:it\:rains=\frac{2}{3}}$

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