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# Introduction to Probability

This tutorial covers material encountered in chapter 13 of the VCE Mathematical Methods Textbook, namely:

• Basic set and probability theory
• Addition and multiplication (Bayes’ Theorem) rules
• Conditional probability
• Law of total probability
• Notion of independent and mutually exclusive events
• Discrete random variables
• Mean, variance and standard deviation of discrete random variables

## Worksheet

Q1. If P(A)=0.3,\,P(B)=0.4 and P(A\cup B)=0.6, are A and B mutually exclusive? Are they independent?

Q2. Show that 1-P(A'\cap B')=P(A\cup B) using visual methods or otherwise. How would you describe this relationship in words to the layman?

Q3. A bag contains three red balls and five green balls. What is the probability of randomly drawing two red balls if:

(a) The first ball drawn is put back in the bag before the second is drawn.

(b) The first ball drawn is not put back in the bag before the second is drawn.

Q4. If P(A)=\dfrac{1}{3},\, P(B)=\dfrac{1}{2} and P(A|B)=\dfrac{1}{4}, find:

(a) P(A\cap B) and P(A\cup B)

(b) P(A'|B) and P(A|B')

Q5. A test for a certain virus is considered 95\% effective; meaning that if someone with the virus is tested the test will return positive 95\% of the time, and if someone without the virus is tested the test will return positive 5\% of the time. Only 1\% of the population actually has the virus. If a person tests positive for the virus what is the probability that they actually have it?

Q6. The discrete random variable X has the following probability distribution:

\begin{array}{c||cccccc}
x & -1 & 0 & 1 & 2 & 3 & 4 \\
\hline P(X=x) & \mathrm{t} & 3 \mathrm{t} & 2 \mathrm{t} & 3 \mathrm{t} & 2 \mathrm{t} & \mathrm{t}
\end{array}

Find:

(a) The constant t

(b) \mu(X)

(c) \sigma(X) and \sigma^2(X)

Q7. A shop manufactures Garfield mugs for \$x. It costs the shop \$3 to make every mug, however there is a \frac{1}{6} chance in manufacturing that the mug will be a defect and in this case it must be thrown out and the mug is considered a total loss to the shop.

(a) Find the probability function P, the profit per Garfield mug.

(b) Find the mean of P in terms of x.

(c) How much should the shop sell its Garfield mugs in order to make a profit in the long term?

### Bonus Questions

#### I. Lewis in probability land

Lewis has a bag with a single ball inside, the ball has an equal probability of being either red or green. Lewis then takes a red ball and puts it inside the bag (with the original ball), shakes it around for a bit and then takes a ball out of the bag at random. If the ball Lewis takes out is red, what is the probability that the other ball was also red?

This question was published in 1895 by Lewis Carroll, the famous author of Alice in Wonderland.

#### II. Monty Hall problem

Monty Hall is the host of a peculiar game show. The contestant is given three doors to choose from, behind one door is \\$10mil and behind the other two doors lies nothing. After the contestant chooses a door (but before they open it) Monty Hall will open a different door that has nothing behind it, he will then give the contestant the choice between switching from their original door to the closed door left over. Should the contestant switch or not?

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