Question

Question 24. 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 tails?

Short answer

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 tails.

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 exactly four heads appear out of flips, and 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

$$\begin{gathered} p\left(\frac{E}{F}\right)=\frac{p(E\cap F)}{P\left(F\right)} \\ =\frac{{\left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^5}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} \\ ={\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^4 \\ =\frac{1}{16} \end{gathered}$$

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