Section 13 - Probability

Learning intentions:

In this section we will examine:

• The fundamental concepts and rules used in the study of probability.
  • How to represent sample spaces for random experiments and events.
  • The addition rule for probability when combining events.
  • Dependent events giving rise to conditional probability.
  • The use of the law of total probability in solving complex problems.
  • Special cases of probability including mutually exclusive events and independent events.
  • The use of simulations, with and without technology, to model probability scenarios.

Please also review set notation from section 1A.

It is also advisable that you are familiar with a deck of cards.

Interesting probability problems:

Presented below are several interesting problems relating to probability and their correct mathematical solutions.

Problem 1:

There are two people, Alan and Bob, playing a gambling game. Each of them pays 30 dollars to start the game (60 dollars in total in the jackpot). Alan and Bob have an equal chance to win, which is 0.5. They make a agreement, whoever is first to win three games gets to take all of the money in the jackpot. Alan wins two games and Bob wins one out of the first three games; however, something happens and the game in interrupted. They are unable to continue playing so they need to know how to divide the money in the jackpot between Alan and Bob.

How do you think it should be done given that Alan won two games and Bob only won once?

Possible solution:

Some people may suggest that as Alan won twice and Bob once the winnings should be split in a 2:1 ratio between Alan and Bob. That is, Alan should get 40 dollars and Bob should get 20 dollars. This sound reasonable, but is it actually fair to Alan and Bob?

Mathematically correct answer:

To answer this question we need to look at the outcomes that would lead to the game being finished. The maximum number of games left are two; therefore, there are four equally likely outcomes:

Of the four possible outcomes (listed above), three result in Alan winning the jackpot while only one results in Bob winning. Therefore, the jackpot should be split in a 3:1 ratio between Alan and Bob. That is, Alan should get 45 dollars and Bob should get 15 dollars. This probability problem is, perhaps, best demonstrated using a tree diagram.

Problem 2:

Mr. and Mrs. Smith have two children, at least one of which is a girl.

What is the probability both of their children are girls?

Possible solution:

Some people may suggest that there is an equal chance of having a boy or a girl each time irrespective of the information given in the question; therefore, the probability is 0.5. This intuitive approach is actually incorrect.

Mathematically correct answer:

To answer this question we need to look at all possible outcomes (the sample space). The possible configurations for the family are shown below:

Of these four options, only three satisfy the condition of having at least one girl. Of these three options, only one results in the family having two girls. Therefore, the probability of the family having two girls is one third (1/3). This probability problem is an example of conditional probability which will be examined in section 13D.

Problem 3:

The Monty-Hall problem is concerned with a counter-intuitive game show problem that can be solved using probability.

Imagine you are on a game show, the game show host shows you three doors. Behind one of the doors is the star prize, a car. Behind the other two doors are booby prizes, two goats. You have no way of knowing which door conceal which item, and whichever door you pick you'll receive the prize behind it. You are asked to pick a door, but before it is opened the game show host opens one of the other two doors. Now, the host knows where the car is and he always opens a door to reveal a goat. You are then asked whether you would like to swap your chosen door for the one remaining closed door.

The question is: Should you swap? Should you stick with your original choice? Or does it make no difference what you do? Which would give you the greatest chance of winning the car?

Mathematically correct answer:

I think this one has been explained perfectly by Ron Clarke in the following YouTube video. Once again, the answer to this problem is not as intuitive as you might think... check it out:

Please rest assured that we will not be asking you to prove the Monty-Hall problem as an assessment task!