Q. 7.70 - A First Course in Probability

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, this value of $p$ varies from coin to coin. Suppose that the composition of the urn is such that if a coin is selected at random from it, then the $p-$value of the coin can be regarded as being the value of a random variable that is uniformly distributed over $\left[0,1\right]$. If a coin is selected at random from the urn and flipped twice, compute the probability that

a. The first flip results in a head; 

b. both flips result in heads. 

Step-by-Step Solution

Answer

a. The first flip results in a head value found to be $P\{$The first flip in a head$\}=\frac{1}{2}$

b. The both flips result in heads value found to be $P\{$Both flips result in heads$\}=\frac{1}{3}$.

Step 1: Given Information (Part a)

If a coin is selected at random from the urn and flipped twice, compute the probability that the first flip results in a head.

Step 2: Explanation (Part a)

The first flip results in a head. 

$P-$value is uniformly distributed over $\left[0,1\right]$,

$P\{$The first flip in a head $\}{\int }_0^1 pdp$.

Step 3: Explanation (Part a)

Substitute the value,

$={\left.\frac{p^2}{2}\right|}_0^1$

Divide the value,

$=\frac{1}{2}$.

Step 4: Final answer (Part a)

The first flip results in a head value found to be $P\{$The first flip in a head $\}=\frac{1}{2}$.

Step 5: Given Information (Part b)

If a coin is selected at random from the urn and flipped twice, compute the probability that the both flips result in heads.  

Step 6: Explanation (Part b)

Both flips result in heads,

The value of a random variable $\left[0,1\right]$ and$2,$

$X-0,1,2$

$P\{$ The first flip in a head$\}{\int }_0^1 dp$.

Step 7: Explanation (Part b)

Substitute the value, 

$={\int }_0^1\left(\begin{array}{l}2\\ 2\end{array}\right)p^2(1-p{)}^{2-2}dp$

$={\left.\frac{p^3}{3}\right|}_0^1$

Divide the value,

$=\frac{1}{3}$.

Step 8: Final answer (Part b)

The both flips result in heads value found to be $P\{$ Both flips result in heads$\}=\frac{1}{3}$.