Activities

Conditional probability - Bayes' theorem type problems

A type of conditional probability problem involves finding the probability of an earlier event having occurred given that we know the outcome for a later event.

This type of problem and similar problems can be solved with the use of Bayes' theorem.

Rather than using Bayes' theorem, we will use the basic formula  and probabilities from a suitable tree diagram.


Consider the example in which two balls are removed from a bag containing three green and two red balls. The first ball is not replaced before the second is removed.

If the second ball is red, what is the probability that the first ball was green?

A = {first ball green} and B = {second ball red}
 

 
 
 

substituting probabilities 
from the tree diagram
There is a simpler approach to this problem:
If the second ball is red, then the first ball must be one of the remaining four balls (three green and one red). Therefore the probability that it is green is 3 out of 4 i.e. 3/4.
 
Other problems do not lend themselves to this simpler treatment. Consider the same bag with the addition of one yellow ball. Again two balls are removed without replacement.

If the balls are different colours, what is the probability that one of the balls is green?

A = {one of the balls is green} and B = {different colours}

Card from pack problem - Two cards are dealt without replacement from a standard pack of 52 cards.  What is the probability that:

Can you use the simpler approach?


Archery problem - Rachel, Susan and Tiffany are shooting at a target in archery. With each arrow, the probability of hitting the centre is:
Rachel , Susan  and Tiffany 
What is the probability that: Solutions