Now that we've had some exposure to Bayesian approaches, let's pause and

think about how these compare to frequentist approaches.

Consider these three statements.

The probability of flipping a coin and getting heads is one-half.

The probability of rolling snake eyes, that is, two 1s on two dice, is 1/36.

The probability of Apple's stock price going up today is 0.75.

What exactly do these statements mean?

How you interpret these statements depends on your definition of probability.

One definition of probability of an event is its

relative frequency in a large number of trials.

For example, if you can repeat flipping a coin indefinitely and

count how many heads you get and

divide that number by the number of flips, the value you obtain should be 0.5.

In other words the probability of event E is defined as the proportion of

times the event occurs and n trials when n goes to infinity.

This is the frequentist definition of probability, suppose now that

you're indifferent between winning a dollar if event E occurs or

winning a dollar if you draw a blue chip from a box with 1,000 x p blue chips and

1,000 x (1-p) white chips.

This means that you're equating the probability of events E, that's P(E),

to the probability of drawing a blue chip from this box.

In other words P(E) = p.

This definition of probability, based on your degree of belief,

is the Bayesian definition.

So, what are the implications of these two different definitions?

In earlier courses in this specialization, we talked about

frequentists methods of inference, for example, a confidence interval.

When defining the confidence level we were very careful to describe it as

the proportion of random samples of size n from the same population

that produced confidence intervals that contain the true population parameter.

We emphasized that an interpretation of the confidence level as the probability

that a given interval containing the true parameter is incorrect.

For example, based on a 2015 Pew Research poll on 1500 adults,

we created the following confidence interval.

We are 95% confident that 60% to 64% of Americans think

the federal government does not do enough for middle class people.

What does 95% confident mean in this statement?

The correct answer is that 95% of random samples of 1,500

adult Americans will produce confidence intervals for the proportion of Americans

who think the federal government does not do enough for a middle class people.