Bayes Theorem

We know that: Given the following Venn Diagram, notice that we can express A as: Therefore: or

An Extension of Conditional Probability Bayes Theorem To use Baye’s Theorem, you need to identify the events A and B. Then you need to find the following probabilities:

A geologist has examined information for an oil well. The geologist reports a 65% chance of finding oil. As drilling begins, sample cores are taken from the well and studied by the geologist. These samples have a history of predicting oil when there is oil about 85% of the time. However, about 6% of the time the sample cores will predict oil when there is no oil. Using these core samples, what is the probability that the well will hit oil? First find events A and B. A: B: the event that the well strikes oil. the event that the core sample indicates oil.

Therefore, using Baye’s Theorem, we have:

A weather satellite is sending a binary code of 0s and 1s describing a storm. Channel noise, though, can be expected to introduce a certain amount of transmission error. Suppose that the message being relayed is 70% 0s and there is an 80% chance of a given 0 or 1 being received properly. If a 1 is received, what is the probability that a 0 was sent? Intro to Math Stats, page 56 Answer: 0.37

A dashboard warning light is supposed to flash red if a car’s oil pressure is too low. On a certain model, the probability of the light flashing when it should is 0.99; 2% of the time, though, it flashes for no apparent reason. If there is a 10% chance that the oil pressure really is low, what is the probability that a driver needs to be concerned if the warning light is on? Intro to Math Stats, page 57 Answer: 0.85

Suppose that 0.5% of all students seeking treatment at a school infirmary are eventually diagnosed as being sick. Of those who are actually sick, 90% complain of a sore throat. But 30% of those not being sick also have sore throats. If a students comes to the infirmary and says that he has a sore throat, what is the probability that he is sick? Intro to Math Stats, page 58 Answer: 0.015

Suppose you have three identical cards, except that both sides of the first card are red, both sides of the second card are black, and one side of the third card is red and the other side black. The three cards are mixed up in a hat and 1 card is randomly selected and put down on a table. If the upper side of the chosen card is red, what is the probability that the other side is colored black? First Course Probability, page 70 Answer: 1/3