Question

Consider an urn containing a large number of coins, and suppose that each of the coins has some probability p of turning up heads when it is flipped. However, p varies from coin to coin. Suppose that the composition of the urn’s contents is such that if a coin is selected at random from it, then the value of p for the coin can be regarded as being the value of a random variable that is uniformly distributed over [0, 1]. If a coin is selected at random from the urn and flipped twice, what is the probability that the first flip results in a head?

Answer 1

Answer:

The required probability is $\frac{1}{2}$

Step-by-step explanation:

Consider the provided information.

We need to find the probability that the first flip results in a head.

Therefore probability of lending head is $P\sim U(1,0)$

The first flip is head can be calculated as:

$\int\limits^1_0 {p} \, dp$

Integrate the above function as shown below:

$\int\limits^1_0 {p} \, dp=[\frac{p^2}{2}]^1_0$

$=\frac{1^2}{2}-\frac{0^2}{2}=\frac{1}{2}$

Hence, the required probability is $\frac{1}{2}$