Question

Question 23. To determine

What is the conditional probability that exactly four heads appear when a fair coin is flipped five times, given that the first flip came up heads?

Short answer

The conditional probability is_{$\frac{1}{16}$}

Step-by-step solution

Step 1. Given information

A coin is flipped five times and the first flip is heads.

Step 2. Definition and formula to be used

Formula used are:

$p\left(\frac{E}{F}\right)=\frac{p(E\cap F)}{P\left(F\right)}$

Step 3. Compute probability

Exactly heads will appear out of the flips if and only if the remaining flips all come up heads. The probability of this is${\left(\frac{1}{2}\right)}^4=\frac{1}{16}$

One can also compute the answer by labelling the event that the first coin flip comes up tails.

Then the probability that a total of heads will appear given that the first flip came up tails is

$$\begin{gathered} p\left(\frac{E}{F}\right)=\frac{p(E\cap F)}{P\left(F\right)} \\ =\frac{{\left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^5}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} \\ =\frac{1}{16} \end{gathered}$$

The conditional probability is $\frac{1}{16}$.