Question

A fair coin is tossed five times. If the out comes are $2$ heads and $3$ tails (in some order), then what is the probability that the fourth toss is a head?

• A. $\frac{1}{4}$
• B. $\frac{2}{5}$
• C. $\frac{1}{2}$
• D. $\frac{3}{5}$

Answer 1

Answer: (B)

$\frac{2}{5}$

A fair coin is tossed $5$ times

$\Rightarrow E=2$ heads are obtained on first two tosses.

$\Rightarrow F=3$ tails are obtained on next three tosses.

Sample space,

$\Rightarrow S=[HHHHH,HHHHT,HHHTH,HHTHH,HTHHH,THHHH,TTHHH,HTTHH,$

$HHTTH,HHHTT,THHHT,HTHTH,THTHH,HHTHT,TTTHH,HTTTH,HHTTT,$

$THHTT,TTHHT,THTHT,HTHTT,TTHTH,TTTTH,TTTHT,TTHTT,THTTT,HTTTT,$

$TTTTT]$

$\Rightarrow$ Sample space $=28$

$\Rightarrow E=2$ heads at first two tosses $=\frac{8}{28}$

$\Rightarrow F=3$ tails on next three tosses $=\frac{4}{28}.$

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

$\Rightarrow P\left(\frac{E}{F}\right)=\frac{\frac{8}{28}}{\frac{4}{28}}$

$=2.$

$\Rightarrow n\left(F\right)=\frac{P\left(\frac{E}{F}\right)}{no.oftosses}$

$=\frac{2}{5}.$

Hence, the answer is $\frac{2}{5}.$