Probability of a probability?

Probability Level 4

You have an unfair coin with probability $p$ of landing on heads. This probability is a random variable uniform in the interval $[0,1]$ .

Now, you flip it and get heads. What is the probability that $p>\frac{1}{2}$ ?

0 1/3 1/2 5/8 2/3 3/4 1 It can't be determined

2 solutions

Calvin Lin Staff
Nov 3, 2016

Let's take the pictorial approach. Since the distribution is uniform, the probability is represented by the corresponding area.

The cases that we land on a H are indicated by the upper right triangle.

The cases where $p > \frac{1}{2}$ are indicated by the green region.

Hence, the probability is $\frac{3}{4}$ .

Geoff Pilling
Nov 3, 2016

Let $H =$ The event that heads is flipped.

Using Baye's Theorem :

$P(p>\frac{1}{2}|H) = \frac{P(H|p>\frac{1}{2})\times P(p>\frac{1}{2})}{P(H)} = \frac{\int_\frac{1}{2}^1 r \, dr}{\int_0^1 r \, dr} = \boxed{\frac{3}{4}}$

Another approach is to consider the pictorial representation, which avoids having to do calculus (but it's essentially the same). I like it because it's easier to see at a glance what is happening.

Just added that as a seperate solution.

Calvin Lin Staff - 4 years, 7 months ago

The pictorial approach is terrible.

Ray C - 4 years, 7 months ago

I like it because it allows you to visually see why the result is $\frac{3}{4}$

Geoff Pilling - 4 years, 7 months ago

Probability of a probability?

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