STROUD Worked examples and exercises are in the text Programme 29: Probability PROGRAMME 29...

STROUD

Worked examples and exercises are in the text

Programme 29: Probability

PROGRAMME 29

PROBABILITY

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Probability

Events and probabilities

Probabilities of combined events

Conditional probability

Probability distributions

Continuous probability distributions

Standard normal curve

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Probability

Random experiments

Events

Sequences of random experiments

Combining events

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Probability

Random experiments

The unknown quantity whose value we are trying to find by performing the experiment is called the result of the experiment and its value, found from an experiment, is called an outcome of the result.

A result can be anticipated but its actual value – the outcome – is unknown until the experiment is completed.

An experiment with a result with more than one possible outcome is referred to as a random experiment.

The outcomes of a random experiment must be mutually exclusive.

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Probability

Events

Whilst the completion of a random experiment will be a single outcome we may not be interested in the specific outcome but whether the outcome lies within a range of possible outcomes.

To cater for ranges of possible outcomes we define an event. An event consists of one or more outcomes selected from a list of all possible outcomes.

An event consisting of a single outcome is called a simple event.

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Probability

Sequences of random experiments

When two or more random experiments are performed one after the other, the final outcome of the sequence of experiments will consist of combinations of the outcomes of the individual experiments.

For example, the two random experiments of tossing a silver coin followed by tossing a copper coin. We shall list the possible outcomes of the first experiment as SH, ST where S stands for silver and of the second as CH, CT where C stands for copper. We can describe this sequence of experiments using an outcome tree.

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Probability

Combining events

Events can be combined using or and and. For example, in the previous sequence of experiments of tossing a silver coin and then tossing a copper coin we could define the events:

D : One silver head or At least one tail E : One silver head and A copper tail

in which case:

D consists of (SH, CH), (SH, CT), (ST, CH) and (ST, CT)

E consists of (SH, CT)

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Probability







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Probability

Assigning probabilities

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Probability

If you tossed a fair coin then there would be 1 chance in 2 of it landing head face uppermost.

If you select six numbers between 1 and 49 for a lottery ticket then there are nearly 14 million possible different selections of six numbers so your selection has a very small chance of winning – in fact it is 1 chance in 13 983 816.

The chance of something happening can be high and can be low but we really want to be more precise than that and quantify chance in a way that makes predicting the future more accurate and more consistent. To do this we use the idea of probability.

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In the random experiment of tossing a coin that has a head on both sides – a double-headed coin the event H is a certain event and we define the probability of a certain event as unity:

P(H) = P(Certainty) = 1

The event T is impossible. We define the probability of an impossible event as zero:

P(T) = P(Impossibility) = 0

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If the double-headed coin is replaced by a normal coin possessing a tail as well as a head then the event H is no longer certain and the event T is no longer impossible. In both cases the events lie somewhere between certainty and impossibility so their probabilities lie somewhere between zero and unity:

0 < P(H) < 1 and 0 < P(T) < 1

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Assigning numbers to these probabilities can be problematic but what we do say is that when a normal coin is tossed it is certain to show either a head or a tail and so the two probabilities must add up to the probability of certainty, that is unity:

P(H) + P(T) = 1

Probabilities can be assigned to the events of a random experiment either beforehand – we call it a priori – or afterwards by statistical regularity.

When we have assigned probabilities to every possible simple event of a random experiment we have what is called a probability distribution.

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Probability







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Or

Non-mutually exclusive events

And

Dependent events

Independent events

Probability trees

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Or

Let A and B be two events associated with a random experiment. These two events can be connected using or to form the event

C: C = A or B

In other words either event A occurs or event B occurs or both. This is an inclusive or because it permits both events to occur simultaneously. If A and B are mutually exclusive they contain no outcomes in common, in which case

P(A or B) = P(A) + P(B)

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Non-mutually exclusive events

If events A and B have outcomes that are common to both then because or is inclusive, when we add together outcomes that are in either A or B we add in twice those outcomes that are in both. Therefore we must subtract once those that are in both. Therefore:

P(A or B) = P(A) + P(B) – P(both A and B)

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And

Two random experiments are performed in sequence where A is an event associated with the first experiment and B an event associated with the second experiment. These two events can be connected via the word and to form the event C where:

C = A and B

That is, both events A and B occur. Furthermore:

P(A and B) = P(A)P(B)

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Dependent events

If two random experiments are performed in sequence, one after the other, then it may be possible for the outcome of the first experiment to affect the outcome of the second experiment. If this is the case then the outcomes are dependent upon each other and the probabilities change after the first experiment has been performed.

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Independent events

If the outcome of the second experiment is unaffected by the outcome of the first experiment then the events are independent of each other and the probabilities will not change after the first experiment has been performed.

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Probability trees

We are already familiar with the idea of a sequence of random experiments and the outcome tree that results from it. If we now list the probabilities against each outcome of the tree we construct what is called a probability tree. For example, in a factory items pass through two processes, namely cleaning and painting. The probability that an item has a cleaning fault is 0.2 and the probability that an item has a painting fault is 0.3. Cleaning and painting faults occur independently of each other so that:

Probability of a cleaning fault P(C)= 0.2Probability of no cleaning fault P(NC) = 0.8 andProbability of a painting fault P(P) = 0.3Probability of no painting fault P(NP) = 0.7

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Probability trees

For example, in a factory items pass through two processes, namely cleaning and painting. The probability that an item has a cleaning fault is 0.2 and the probability that an item has a painting fault is 0.3. Cleaning and painting faults occur independently of each other so that:

Probability of a cleaning fault P(C) = 0.2Probability of no cleaning fault P(NC) = 0.8

and

Probability of a painting fault P(P) = 0.3Probability of no painting fault P(NP) = 0.7

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Probability trees

This gives rise to the following probability tree:

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Probability







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We are concerned here with the probability of an event B occurring, given that an event A has already taken place. This is denoted by the symbol P(B|A). If A and B are independent events, the fact that event A has already occurred will not affect the probability of event B. In that case:

P(B|A) = P(B)

If A and B are dependent events, then event A having occurred will affect the probability of the occurrence of event B.

If A and B are independent events: P(A and B) = P(A)P(B)

If A and B are dependent events: P(A and B) = P(A)P(B|A)

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Random variables

Expectation

Variance and standard deviation

Bernoulli trials

Binomial probability distribution

Expectation and standard variation

The Poisson probability distribution

Binomial and Poisson compared

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Random variables

Every random experiment gives rise to a collection of mutually exclusive outcomes, each with an associated probability of that outcome occurring. This collection of probabilities is called the probability distribution of the random experiment and we have seen how a probability distribution can be divined either from a relative frequency distribution using the notion of statistical regularity or from a priori considerations.

Whichever method we choose to create the probability distribution the process can be greatly assisted with the notion of a random variable x that is created by coding each outcome with a number.

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Expectation

Every permitted value of a random variable x associated with a random experiment has a probability P(x) of being realized. In analogy with defining the average value of a collection of data as the sum of the product of each datum value with its relative frequency, we can define the average value of a random variable as the sum of the product of each of its values with its respective probability:

Here, the average value of the random variable is called the expectation of x, denoted by E(x). That is:

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Variance and standard deviation

The spread of the values of the random variable about the mean (the expectation) is given as the expectation of the square of the deviations from the mean (the variance). That is

Where is the standard deviation

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Bernoulli trials

A Bernoulli trial is any random experiment whose result has only two outcomes which we shall call success with probability p and failure with probability q.

P(success) = p and P(failure) = q where p + q = 1

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Binomial probability distribution

The binomial probability distribution is concerned with the probability of r successes in n Bernoulli trials and is given by:

p is the probability of success, q is the probability of failure and p + q = 1.

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Expectation and standard variation

The expectation of a binomial probability distribution is given as:

The standard deviation of a binomial probability distribution is given as:

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The Poisson probability distribution

If we know the average number of occurrences of an event during a fixed period of time then the Poisson probability distribution will enable us to compute the probabilities of 0, 1, 2, 3, ... occurrences during that same interval of time. The probabilities are given by:

with mean and variance both equal to λ (so standard deviation is √λ ). Here λ is the average number of occurrences during a fixed period of time and r is a positive integer.

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Binomial and Poisson compared

If the mean of the Poisson probability distribution λ is less than 5, the probabilities obtained from the Poisson distribution are a good approximation to those obtained using the binomial probability distribution, particularly if the number of trials n is large (n ≥ 50) and the probability of success p is small (p ≤ 0:1); what are called rare events. For this reason, it can be more convenient to calculate probabilities using the Poisson distribution because the calculations involved are more easily performed. In such a case we take λ = μ where μ = np, the mean of the binomial probability distribution.

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Probability







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Normal distribution curve

The equation of the normal curve is

Where μ = mean and σ = standard deviation of the distribution.

This equation is not at all easy to deal with. In practice, it is convenient to convert a normal distribution into a standardized normal distribution having a mean of 0 and a standard deviation of 1.

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The conversion from normal distribution to standard normal distribution is achieved by the substitution of the standard normal variable z where

This moves the distribution curve along the x-axis and reduces the scale of the horizontal units by dividing by σ. To keep the total area under the curve at unity, we multiply the y-values by σ. The equation of the standardized normal curve then becomes:

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Programme 29: ProbabilityLearning outcomes

Understand what is meant by a random experiment Distinguish between the result and an outcome of a random experiment Recognize that, whilst outcomes are mutually exclusive, events may not be Combine events and construct an outcome tree for a sequence of random experiments Assign probabilities to events and distinguish between a priori and statistical

regularity Distinguish between mutually exclusive and non-mutually exclusive events and

compute their probabilities Distinguish between dependent and independent events and apply the multiplication

law of probabilities Compute conditional probabilities Use the binomial and Poisson probability distributions to calculate probabilities Use

the standard normal probability distribution to calculate probabilities

STROUD Worked examples and exercises are in the text Programme 29: Probability PROGRAMME 29...

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Transcript of STROUD Worked examples and exercises are in the text Programme 29: Probability PROGRAMME 29...