Let $P\left(A\right)=\frac{1}{4},P\left(A^{\prime }\right)=\frac{3}{4},P\left(B\mid A\right)=\frac{3}{5},$ and $P\left(B\mid A^{\prime }\right)=\frac{1}{2}$. What is $P\left(A\mid B\right)$ according to Bayes’ Theorem?

Given $P\left(E\right)=0.47$, $P\left(E^{\prime }\right)=0.53$, $P\left(F\mid E\right)=0.29$, and $P\left(F\mid E^{\prime }\right)=0.11$, use Bayes’ Theorem to find $P\left(E\mid F\right)$.

Let $P\left(G\right)=0.5,P\left(G^{\prime }\right)=0.5,P\left(H\mid G\right)=0.4,$ and $P\left(H\mid G^{\prime }\right)=0.1$. What is $P\left(G\mid H\right)$ according to Bayes’ Theorem?

A survey of drivers recorded whether each driver texted while driving and whether they received a warning last year. The counts from the survey show that $30$ drivers both texted while driving and received a warning, and $120$ drivers received a warning in total. What is the probability that a driver texts while driving, given that they received a warning last year?

A university study reports: $55\%$ of students who complete a tutoring program pass the final exam; $35\%$ of students who complete the tutoring program and attend all review sessions pass the final exam; and $10\%$ of students who complete the tutoring program, attend all review sessions, and submit all assignments pass the final exam. Which expression correctly represents the statement "$35\%$ of students who complete the tutoring program and attend all review sessions pass the final exam" as a conditional probability?

A certain disease is present in $2\%$ of a population. A screening test for the disease is positive $85\%$ of the time when the person has the disease and $7\%$ of the time when the person does not have the disease. Let $E$ be the event "the person has the disease" and $F$ be the event "the test is positive." What is the probability that a person actually has the disease given that they test positive?

Given $P\left(E\right)=0.3,P\left(E^{\prime }\right)=0.7,P\left(F\mid E\right)=0.6$, and $P\left(F\mid E^{\prime }\right)=0.2$, use Bayes’ Theorem to find $P\left(E\mid F\right)$.