# One-Stage Probability In Gambling

How often do you hear people say things like "I *never* win the lottery", "It's *likely* to rain today",
"There's a *50:50* chance that our team will win the championship", "If he speeds, he'll *probably* get caught by the police" and "She *always* tells the truth"?

All of these comments are in some way related to probability (chance).

The **Probability** of an event happening is represented from **0 (Never Happens) to 1 (Always happens)**.
Probabilities may be written as **simplified fractions** (e.g. ^{1}⁄_{2} ), **decimals** or **relative frequencies** (e.g. 0.5), or **percentages** (e.g. 50%).

## Questions

What **Probability from 0 to 1** would each of the Following Events Be?

**Q1.** If you flip a coin in the air, it will land with the head's side upwards.
**Q2.** If you throw a 6-sided die (dice), it will land with the 3 upwards.
**Q3.** Today is your birthday.
**Q4.** The Sun will shine tomorrow.
**Q5.** An earthquake will shake your house this year.
**Q6.** You will suffer from pteromerhanophobia if you fly with the Australian airline QANTAS.

**Answers:**

**A1.** Because the coin has 2 equal sides, the probability will be ^{1}⁄_{2} or 0.5.

**A2.** The side with the 3 is only one of the 6 equal sides. The Probability of any side landing upwards is ^{1}⁄_{6} or 0.166666.

**A3.** In a year with 365 days, the chance of today being your birthday is ^{1}⁄_{365} or 0.00274. If this year is a leap year with 366 days, the chance is a slightly less ^{1}⁄_{366} or 0.00273.

**A4.** One. Astronomers believe that the Sun is about 4.5 billion years old and is about halfway through its life cycle. It is extremely improbably that the Sun will not shine today. However, if the Sun did explode NOW, you and I would not see it for another 8.3 minutes because this is the time it takes for sunlight to travel from the Sun to the Earth.

**A5.** Quakes are surprisingly common and this also depends on where you live. The probability is probably more than 0.5 if you live near in a quake area. If not, the chance may be closer to zero.

**A6.** Close to zero. Pteromerhanophobia is a fear of flying. QANTAS is the only world airline that has *never* had a fatal jet accident.

## Probability of Tossing a Coin

**Fair Coin** - A fair coin is one which has 2 equal sides, is equally weighted on each side, and has the same chance of landing on each side.

**Sample Space** - This is all the possible **outcomes** that can occur. When tossing a coin, the sample space is one head and one tail which is written as **H, T**

Probability (a **Head** in one coin toss) = ^{1}⁄_{2} = 0.5

Probability (a **Tail** in one coin toss) = ^{1}⁄_{2} = 0.5

**When you add all the Probabilities, they add up to 1**.

## Probability of Tossing a Six-Sided Die

**Die** - This word is the singular.
**Dice** - This word is the plural. It's a little like "mouse" and "mice".

**Fair Die** - A fair die is one which has 6 equal sides, is equally weighted on each side, and has the same chance of landing on each side.

**Sample Space** - This is all the possible **outcomes** that can occur. When tossing a die, the sample space is **1, 2, 3, 4, 5, 6**.

Probability (a **1** in one die toss) = ^{1}⁄_{6}

Probability (a **2** in one die toss) = ^{1}⁄_{6}

Probability (a **3** in one die toss) = ^{1}⁄_{6}

Probability (a **4** in one die toss) = ^{1}⁄_{6}

Probability (a **5** in one die toss) = ^{1}⁄_{6}

Probability (a **6** in one die toss) = ^{1}⁄_{6}

**When you add all the Probabilities, they add up to 1**.

## Example One

What is the probability of throwing an even number on a die?

**Answer:**

The even-numbered sides are 2, 4, 6. This is 3 sides out of 6 sides.

P (even number) = ^{3}⁄_{6} = ^{1}⁄_{2}

## Example Two

What is the probability of throwing a number more than 2 on a die?

**Answer:**

The numbers more than 2 are 3, 4, 5, 6. This is 4 side out of 6 sides.

P (number more than 2) = ^{4}⁄_{6} = ^{2}⁄_{3}

## Example Three

In a casino game, you will win money if you throw a 3 *or* an even number on a die. What is the chance of this?

**Answer:**

This means that you will win if you throw 3 or 2, 4, 6. This is 4 sides out of 6 sides.

P (3 or even number) = ^{4}⁄_{6} = ^{2}⁄_{3}

## Example Four

In a casino game, you will win money if you throw a 3 *and* a 6 on a die. What is the chance of this?

**Answer:**

It is impossible to throw both a 3 *and* a 4 on one toss of a die.

P (3 and 4) = 0

## Did You Know That...?

In 2009, Austrian home-owner Traude Daniel had difficulty selling her luxury million-dollar home so she raffled it. There were 9999 tickets sold at $128 each. What is the chance of winning this lottery? Did Traude make enough money?

## Probability with 52 Playing Cards

In mathematical questions about playing cards, it is assumed that the pack consists of 52 playing cards with no jokers. The pack consists of:

**2 Colours**- Half of the cards are**Red**(**Hearts ♥**and**Diamonds ♦**) and the other half are**Black**(**Spades ♠**and**Clubs ♣**).**4 Suits**- One quarter of the pack is**Hearts ♥**, one quarter is**Diamonds ♦**, one quarter is**Spades ♠**, and the other quarter is**Clubs ♣**.**13 Cards in Each Suit**- Each of the 4 suits has the following cards - Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King.

## Example Five

If I choose one card from a pack of 52 cards, what is the probability of each of the following events occurring?

P (a red card) = ^{26}⁄_{52} = ^{1}⁄_{2}

P (a black card) = ^{26}⁄_{52} = ^{1}⁄_{2}

P (a heart ♥) = ^{13}⁄_{52} = ^{1}⁄_{4}

P (a diamond ♦) = ^{13}⁄_{52} = ^{1}⁄_{4}

P (a spade ♠) = ^{13}⁄_{52} = ^{1}⁄_{4}

P (a club ♣) = ^{13}⁄_{52} = ^{1}⁄_{4}

P (an Ace) = ^{4}⁄_{52} = ^{1}⁄_{13}

P (2) = ^{4}⁄_{52} = ^{1}⁄_{13}

P (3) = ^{4}⁄_{52} = ^{1}⁄_{13}

P (a King) = ^{4}⁄_{52} = ^{1}⁄_{13}

P (the Ace of Hearts) = ^{1}⁄_{52}

P (the 7 of Clubs) = ^{1}⁄_{52}

**Careful with these!**

P (an Ace or a King) = ^{4}⁄_{52} + ^{4}⁄_{52} = ^{8}⁄_{52} = ^{4}⁄_{13}

P ( an Ace or a Heart) = ^{4}⁄_{52} + ^{13}⁄_{52} – ^{1}⁄_{52} = ^{16}⁄_{52} = ^{4}⁄_{13} *(Be careful not to count the Ace of Hearts twice!)*

P (an Ace and a King) = 0 *(Impossible!)*

## Questions - P.R.O.B.A.B.I.L.I.T.Y.

In the word PROBABILITY, you are to choose one letter. What is the probability that the letter is:

**Q1.** the first letter of the alphabet (A)
**Q2.** the second letter of the alphabet (B)
**Q3.** the last letter of the alphabet (Z)
**Q4.** a vowel (A,E,I,O,U)
**Q5.** not a vowel ?

**Answers:**

**A1.** ^{1}⁄_{11}

**A2.** ^{2}⁄_{11}

**A3.** 0

**A4.** ^{4}⁄_{11}

**A5.** ^{7}⁄_{11}

## Example Six - Tossing a Coin Expectation

If a fair coin is tossed 200 times, how many heads would be expected to occur?

**Answer:**

P (a Head in one coin toss) = ^{1}⁄_{2}

Expected numbers of heads in 200 tosses = ^{1}⁄_{2} × 200 = 100 heads

## Example Seven - Tossing a Die Expectation

If a die is thrown 30 times, how many times would a 6 be expected to occur?

**Answer:**

P (a 6 in one die toss) = ^{1}⁄_{6}

Expected numbers of heads in 30 tosses = ^{1}⁄_{6} × 30 = 5 times

## Question - Expectation

In a packet of jelly beans, ^{1}⁄_{5} are red (my favourite). If I eat a handful of 10 jelly beans (without looking), how many of my favourite red ones should I get?

**Answer:**

2