Probability

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The probability of an outcome is defined as

$$\mathrm{probability}=\frac{\mathrm{number\; of\; successful\; outcomes}}{\mathrm{total\; number\; of\; possible\; outcomes}}$$

This can be written using algebra as

$$P(X=x)=\frac{N(X=x)}{n}$$

As $0\le N(X=x)\le n$ then $0\le P(X=x)\le 1$.

If the probability of an event is $P$ then the probability of it not happening is $1-P$

If all outcomes are included in the list of events then the list is exhaustive.

The probabilities of an exhaustive list of mutually exclusive events sum to one.

$$P(X=x)$$

This is read as the probability, $P$, of an event, $X$, being equal to an outcome, $x$, of the sample space.

Predicted number of outcomes = probability × number of trials

$$n(X=x)=NP(X=x)$$

A sample space diagram shows all the possible outcomes. You can use it to find the theoretical probability.

You can estimate the probability of an event from the results of an experiment or survey

$$\mathrm{estimated\; probability}=\frac{\mathrm{frequency\; of\; an\; event}}{\mathrm{total\; frequency}}$$

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The Addition Rule

If two events, A and B, are not mutually exclusive then the probability that A or B will occur is given by the addition formula:

$$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$$

Don't panic, this just means: The probability of $A$ or $B$ occurring is the probability of $A$ add the probability of $B$ minus the probability that they both occur. This is best seen by an example...

Example:

Pick a card at random from a pack of 52 cards. Find the probability that you pick an ace or a spade.

We want P(ace or spade) or P(ace ∪ spade).

P(ace ∪ spade) 	= P(ace) + P(spade) − P(ace ∩ spade)
= 4/52 + 13/52 − 1/52
= 16/52 or 4/13

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Mutually Exculsive Events:

Mutually exclusive events are those that cannot happen at the same time.

For mutually exclusive events, on a Venn diagram ther is no intersection, so $P(A\cap B)=0$ and the addtion rule becomes $P(AorB)=P\left(A\right)+P\left(B\right)$;

More ...

Mutually Exclusive Events - GCSE Maths - Steps & Examples
(Care by thirdspacelearning.com)

Mutually Exclusive Events (Care by www.mathsisfun.com)

Mutually Exclusive Events and Independence (Care by mmerevise.co.uk)

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Remember:

$$P(\mathrm{ace}\cap \mathrm{spade})=P(\mathrm{ace}and\mathrm{spade})$$

= 4/52 × 13/52
= 1/52 i.e. the card is the ace of spades

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1. Probabilities and frequencies of events can be shown using a Venn diagram;
2. For independent events, $P(AandB)=P\left(A\right)\times P\left(B\right)$;
3. Two or more events can also be represented by a tree diagram.

(taken from Edexcel A Level year one syllabus )

Probability: basics Video

Probability: not happening Video

Probability: scale Video

Probability: OR rule Video

Practice Questions

Textbook Exercise

Probability: listing outcomes Video

Practice Questions

Textbook Exercise

Probability: sample space Video

Practice Questions

Textbook Exercise

Probability: relative frequency Video

Practice Questions

Textbook Exercise

Probability: independent events Video

Practice Questions

Textbook Exercise

Probability Websites:

• Probability mass function ;
• Discrete Random Variables;
• Pascal's Triangle;
• Quincunx;

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Tree Diagrams

Probability: tree diagrams Video

Practice Questions [inc MS ];

Textbook Exercise [ MS ];

(Care of corbettmaths.com)

Probability Tree Diagrams.

(Care of studywell.com)

Probability Trees

Exam Questions

Probability Trees

Solutions

(Care of www.mathsgenie.co.uk)

Tree Diagrams idependant

(Care of mrbartonmaths.com)

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Distributions

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Binomial Distribution

Warm up:

Binomial Expansion

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Description:

$${(p+q)}^n=\underset{r=0}{\overset{n}{\Sigma }}\left(\begin{array}{c}n\\ r\end{array}\right)p^r{q}^{n-r}$$

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Normal Distribution

68–95–99.7 rule

$$P(\mu -\sigma \le X\le \mu +\sigma )\approx 68.27\%$$ $$P(\mu -2\sigma \le X\le \mu +2\sigma )\approx 95.45\%$$ $$P(\mu -3\sigma \le X\le \mu +3\sigma )\approx 99.73\%$$

These ideas can be useful in two ways:

to confirm that a random variable is normally distributed;

to identify the expected frequency of with these ranges for a random variable of known mean and standard deviation.

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Conditional probability

Multiplication Rule for Probability

$$P(A\cap B)=P\left(A\right|B)\times P(B)$$

$P\left(A\right|B)$ is read as "the probability of A given B"

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Independent or Conditional Probability

Events can be either independent or conditional. In independent events the results of the first event has no bearing on the results of the second event. In conditional events, the outcome of the first event can change the probability of a following event. An example is the probability of your being late for work will change depending on whether you overslept. Situations like this can be modelled using conditional probability. You use a vertical line to indicate conditional probabilities.

• The probability that $B$ occurs given that $A$ has already occurred is $P\left(B\right|A)$. Similarly, $P\left(B\right|A^{\text{'}})$ describes the probability of $B$ occurring given that $A$ has not occurred;
• For independent events, $P\left(A\right|B)=P(A\left|B^{\text{'}}\right)=P\left(A\right)$ and $P\left(B\right|A)=P(B\left|A^{\text{'}}\right)=P\left(B\right)$.

You can use this condition to determine independence. You can solve some problems involving conditional probability by considering a restricted sample space of the outcomes where one event has already occurred.

You can find conditional probabilities from a Venn diagram by considering the section of the Venn diagram that corresponds to the restricted sample space.

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Probability: conditional Video

Practice Questions

Textbook Exercise

(Care of corbettmaths.com)

conditional-probability-and-tree-diagrams (Care of mmerevise.co.uk)

Probability (Care of revisionmaths.com/)

Set notation and conditional probability (Care of www.savemyexams.com)

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