Feel free to try out your own die rolling strings, like '5d4 + 3' or '6d8'. The 'd' must be lowercase. To drop the lowest x dice, use an uppercase D, like '4d6D1' means to roll 4d6 and drop the lowest 1 die. Craps Check: Betting on any craps during the come out roll to hedge your pass line bet. Don't Come bet: A don't pass bet made after the point is established. Don't Pass bet: A bet that the shooter will not make his point. Double odds: An odds bet that is twice the size of the original pass/come bet. Some casinos offer higher odds.

1. Wahrscheinlichkeit Craps Play
2. Wahrscheinlichkeit Craps App
3. Wahrscheinlichkeit Craps Odds
4. Wahrscheinlichkeit Craps Games
5. Wahrscheinlichkeit Craps Table

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How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin

When a coin is tossed, there are two possible outcomes:

• heads (H) or
• tails (T)

We say that the probability of the coin landing H is ½

And the probability of the coin landing T is ½

Throwing Dice

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6.

The probability of any one of them is 16

Probability

In general:

Probability of an event happening = Number of ways it can happenTotal number of outcomes

Example: the chances of rolling a '4' with a die

Number of ways it can happen: 1 (there is only 1 face with a '4' on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 16

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 45 = 0.8

Probability Line

We can show probability on a Probability Line:

Probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads.

But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Learn more at Probability Index.

Words

Some words have special meaning in Probability:

Experiment: a repeatable procedure with a set of possible results.

Example: Throwing dice

We can throw the dice again and again, so it is repeatable.

The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}

Sample Space: all the possible outcomes of an experiment.

Example: choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards: {Ace of Hearts, 2 of Hearts, etc... }

The Sample Space is made up of Sample Points:

Example: Deck of Cards

• the 5 of Clubs is a sample point
• the King of Hearts is a sample point

'King' is not a sample point. There are 4 Kings, so that is 4 different sample points.

Example: Throwing dice

There are 6 different sample points in the sample space.

Example Events:

Wahrscheinlichkeit Craps Play

An event can be just one outcome:

• Getting a Tail when tossing a coin
• Rolling a '5'

An event can include more than one outcome:

• Choosing a 'King' from a deck of cards (any of the 4 Kings)
• Rolling an 'even number' (2, 4 or 6)

Hey, let's use those words, so you get used to them:

Example: Alex wants to see how many times a 'double' comes up when throwing 2 dice.

Wahrscheinlichkeit Craps App

The Sample Space is all possible Outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ... {6,3} {6,4} {6,5} {6,6}

The Event Alex is looking for is a 'double', where both dice have the same number. It is made up of these 6 Sample Points:

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

Wahrscheinlichkeit Craps Odds

These are Alex's Results:

Wahrscheinlichkeit Craps Games

After 100 Experiments, Alex has 19 'double' Events ... is that close to what you would expect?

Wahrscheinlichkeit Craps Table