Question A

State the following probabilities:

The probability of getting a heads when tossing a fair coin.
The probability of drawing a red card in a shuffled deck.
The probability of drawing the eight of clubs in a shuffled deck.
The probability of rolling snake’s eyes (two 1’s) when rolling a pair of dice.
The probability of drawing an even heart or and odd club in a shuffled deck.
The probability winning a four ball lottery (balls numbered from 1 to 59).

Question B

Three drugs are to be tested on a group of 500 people, they are administered randomly as follows:

On the first month half the population is chosen randomly to be given drug A.
On the second month 130 people are chosen randomly to be given drug B.
On the third month 40% of those people not given drug A are chosen randomly to be given drug C.

Draw a tree diagram of this.
What is the probability of being given drug C?
What is the probability of being given both drug A and B?
What is the probability of not receiving any drug at all?

Question C

Two events \(A\) and \(B\) have probability \(\mathbb{P}(A) = \frac{1}{6}\) and \(\mathbb{P}(B) = \frac{1}{2}\) of occurring, and \(\mathbb{P}(A \cap B) = \frac{2}{15}\).

Find \(\mathbb{P}\left(\overline{A}\right)\)
Find \(\mathbb{P}(A \cup B )\)
Find \(\mathbb{P}\left(\overline{A} \cap B\right)\)
Find the probability of \(A\) or \(B\) occurring, but not both.

Question D

In a certain card game only the Diamonds suit used (13 cards). You are dealt 2 cards. What is the probability of:

Getting a King?
Getting a Queen and a Jack?
Getting two cards that sum to 15? (face cards are not worth any points, aces are worth 1 point)

Question E

The government wants to know the proportion of people who have ever committed adultery. As this may be a potentially dangerous question to answer truthfully, the government anonymises the answer in the following manner: Participants are asked to roll two fair die, if both numbers are even must answer truthfully, otherwise they must lie. The die rolls are not recorded, so in this way there is no way for the anyone to find out who is lying. From 4000 replies, 2360 people say they have committed adultery. What is the true proportion of people who have committed adultery?

Question F

Show whether the following pairs of events are independent or not:

“Drawing a heart” and “Drawing a red” from a shuffled deck of 52 cards (no jokers).
“Drawing an 8” and “Drawing a spade” from a shuffled deck of 52 cards (no jokers).

Question G

In New York 46% of all teenagers own a skateboard and 36% of all teenagers own both a skateboard and roller blades. What is the probability that a teenager owns roller blades given that the teenager owns a skateboard?

Question H

The most popular first name in the world is Muhammed. The most popular middle name in the word is James. The most popular surname in the word is Wang.

Explain why it would be uncommon to meet someone named Muhammed James Wang.

Question I

Consider three events \(X\), \(Y\) and \(Z\). It is known than \(X\) and \(Y\) are mutually exclusive, \(\mathbb{P}(X) = 0.25\), \(\mathbb{P}(Y) = 0.25\), and \(\mathbb{P}(Z) = 0.4\). It is also known than \(\mathbb{P}(X \cap Z) = 0.05\) and \(\mathbb{P}(Y \cap Z) = 0.01\).

Find \(\mathbb{P}(X \cap Y)\)
Find \(\mathbb{P}(X \cup Y)\)
Find \(\mathbb{P}(X \cap Y \cap Z)\)
Find \(\mathbb{P}\left(X \cap \overline{Z}\right)\)
Find \(\mathbb{P}\left(\overline{X} \cap \overline{Y} \cap \overline{Z}\right)\)
Find \(\mathbb{P}(X \mid Z)\)
Find \(\mathbb{P}(Z \mid X)\)
Find \(\mathbb{P}(X \cup Y \cup Z)\)

Question J

The Pokérus virus is rare, it affects 1 in 21,845 Pokémon. Nurse Joy can test for the virus, but only with 98% accuracy. Brock suspects his Geodude had Pokérus, and the test is positive. What is the probability that Brock’s Geodude actually has the Pokérus virus?

(Bonus - What assumptions have you made that are not met by the wording of the question?)

Question K

Here is a well known problem: the problem of two liars: Andy and Beatrice both only tell the truth a third of the time. A wallet is stolen and Andy says that he did not steal the wallet. Beatrice then confirms this. What is the probability that Andy is actually telling the truth?