# Exercises - Bayes' Theorem

1. In Orange County, 51% of the adults are males. One adult is randomly selected for a survey involving credit card usage. It is later learned that the selected survey subject was smoking a cigar. Also, 9.5% of males smoke cigars, whereas 1.7% of females smoke cigars (based on data from the Substance Abuse and Mental Health Services Administration). Use this additional information to find the probability that the selected subject is a male.

0.853

2. A person uses his car 30% of the time, walks 30% of the time and rides the bus 40% of the time as he goes to work. He is late 10% of the time when he walks; he is late 3% of the time when he drives; and he is late 7% of the time he takes the bus.

1. What is the probability he took the bus if he was late?
2. What is the probability he walked if he is on time?
a) 0.418; b) 0.289

3. In a study of pleas and prison sentences, it is found that 45% of the subjects studied were sent to prison. Among those sent to prison, 40% chose to plead guilty. Among those not sent to prison, 55% chose to plead guilty.

1. If one of the study subjects is randomly selected, find the probability of getting someone who was not sent to prison.
2. If a study subject is randomly selected and it is then found that the subject entered a guilty plea, find the probability that this person was not sent to prison.
a) 0.55; b) 0.627

4. On a game show, a contestant can select one of four boxes. The red box contains one $\$100$bill and nine$\$1$ bills. A green box contains two $\$100$bills and eight$\$1$ bills. A blue box contains three $\$100$bills and seven$\$1$ bills. A yellow box contains five $\$100$bills and five$\$1$ bills. The contestant selects a box at random and selects a bill from the box at random. If a $\$100\$ bill is selected, find the probability that it came from the yellow box.

0.455