The toss the a coin, throwing dice and also lottery draws room all examples of arbitrarily events.

You are watching: Probability of 3 heads in a row

There can be:

**Dependent Events**where what happens

**depends on**what taken place before, such as acquisition cards native a deck makes less cards each time (learn an ext at Conditional Probability), or

## Independent Events

Independent occasions are **not affected** by ahead events.

**This is an important idea!**

A coin does no "know" it come up heads before.

And every toss that a coin is a perfect isolated thing.

### Example: girlfriend toss a coin and also it comes up "Heads" 3 times ... What is the chance that **the next toss** will also be a "Head"?

**The opportunity is simply ½ (or 0.5) simply like any toss of the coin.**

What the did in the previous will not influence the present toss!

Some world think "it is overdue for a Tail", but really important the following toss that the coin is completely independent of any type of previous tosses.

Saying "a Tail is due", or "just one more go, mine luck is due to change"is called **The Gambler"s Fallacy**

Of course her luck **may** change, since each toss the the coin has actually an same chance.

## Probability of elevation Events

"Probability" (or "Chance") is **how likely** miscellaneous is come happen.

So just how do us calculate probability?

### Example: what is the probability of getting a "Head" as soon as tossing a coin?

**Number of ways it deserve to happen: 1** (Head)

**Total number of outcomes: 2** (Head and also Tail)

So the probability = *1***2** = 0.5

### Example: what is the probability of obtaining a "4" or "6" once rolling a die?

**Number of ways it can happen: 2** ("4" and also "6")

**Total number of outcomes: 6** ("1", "2", "3", "4", "5" and "6")

So the probability = *2***6** = *1***3** = 0.333...

## Ways of mirroring Probability

Probability goes indigenous **0** (imposssible) to **1** (certain):

It is often displayed as a **decimal** or **fraction**.

### Example: the probability of acquiring a "Head" once tossing a coin:

As a decimal:**0.5**As a fraction:

**1/2**As a percentage:

**50%**Or occasionally like this:

**1-in-2**

## Two or more Events

We deserve to calculate the chances of two or much more **independent** events by **multiplying** the chances.

### Example: Probability that 3 top in a Row

For each toss of a coin a Head has actually a probability of 0.5:

And so the chance of obtaining 3 heads in a row is **0.125**

So every toss of a coin has a ½ opportunity of gift Heads, but** several Heads in a row** is unlikely.

### Example: Why is it i can not qualify to get, say, 7 top in a row, when each toss of a coin has actually a ½ opportunity of being Heads?

Because we space asking two various questions:

Question 1: What is the probability of 7 heads in a row?

Question 2: as soon as **we have actually just acquired 6 heads** in a row, what is the probability the **the following toss** is likewise a head?

You have the right to have a play with the Quincunx come see just how lots of independent impacts can still have a pattern.

## Notation

We use "P" to average "Probability Of",

So, because that Independent Events:

P(A and also B) = P(A) × P(B)

Probability that A and also B equals the probability that A time the probability of B

### Example: your boss (to it is in fair) randomly assigns anyone an extra 2 hours work ~ above weekend evenings in between 4 and midnight.

What space the possibilities you gain Saturday in between 4 and also 6?

**Day:** there are two work on the weekend, so **P(Saturday) = 0.5**

**Time:** you desire the **2 hours** the "4 come 6", out of the **8 hours** that 4 to midnight):

**P("4 come 6") = 2/8 = 0.25**

And:

P(Saturday and also "4 come 6") | = P(Saturday) × P("4 to 6") |

= 0.5 × 0.25 | |

= 0.125 |

Or a 12.5% Chance

(Note: us could also have settled that you wanted 2 hours out of a total possible 16 hours, i m sorry is 2/16 = 0.125. Both approaches work here.)

## Another Example

### Example: the chance of a flight being delayed is 0.2 (=20%), what room the opportunities of no delays ~ above a round trip

The possibility of a flight **not** having a hold-up is 1 − 0.2 = **0.8**, for this reason these room all the possible outcomes:

0.8 × 0.8 = | 0.64 chance of no delays | |

0.2 × 0.8 = | 0.16 chance of first flight delayed | |

0.8 × 0.2 = | 0.16 opportunity of return flight delayed | |

0.2 × 0.2 = | 0.04 chance of both flights delayed |

When we include all the possibilities we get:

0.64 + 0.16 + 0.16 + 0.04 = **1.0**

They all add to 1.0, which is a great way of checking our calculations.

Result: **0.64**, or a 64% possibility of no delays

## One an ext Example

Imagine there are two groups:

A member the each team gets randomly favored for the winners circle,**then**among those it s okay randomly liked to gain the large money prize:

What is your opportunity of winnning the big prize?

there is a**1/5 chance**of going to the winners circle and also a

**1/2 chance**of to win the large prize

So you have a 1/5 chance followed by a 1/2 opportunity ... Which provides a 1/10 opportunity overall:

*1***5** × *1***2** = *1***5 × 2** = *1***10**

Or we have the right to calculate making use of decimals (1/5 is 0.2, and 1/2 is 0.5):

0.2 x 0.5 = **0.1**

So your chance of to win the huge money is **0.1** (which is the exact same as 1/10).

## Coincidence!

Many "Coincidences" are, in fact, likely.

### Example: you space in a room with 30 people, and also find that Zach and also Anna celebrate your birthday ~ above the very same day.

Do you say:

"Wow, just how strange !", or "That seems reasonable, with so many civilization here"In reality there is a **70% chance** that would take place ... So the is **likely**.

Why is the possibility so high?

Because you are comparing anyone to anyone else (not just one to many).

And with 30 world that is **435 comparisons**

(Read shared Birthdays to find out more.)

### Example: Snap!

Did you ever before say other at precisely **the exact same time as someone else**?

Wow, just how amazing!

**But** girlfriend were more than likely sharing an suffer (movie, journey, whatever) and also so your thoughts were similar.

And there are only so numerous ways of saying something ...

... So it is favor the card game "Snap!" (also called Slaps or Slapjack) ...

... If girlfriend speak enough words together, they will eventually match up.

So, maybe not therefore amazing, just simple chance at work.

See more: How Many Volts Does A Microwave Use, Section F: Watts, Volts, And Amps, Oh My!

Can girlfriend think that other situations where a "coincidence" was simply a most likely thing?

## Conclusion

Probability is: (Number of methods it deserve to happen) / (Total variety of outcomes)Dependent occasions (such as removing marbles native a bag) are impacted by ahead eventsIndependent events (such as a coin toss) room

**not**impacted by ahead eventsWe can calculate the probability of two or much more

**Independent**occasions by

**multiplying**Not every coincidences room really unlikely (when girlfriend think about them).

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