Question

A fair coin is tossed $8$ times find the probability that it shows heads
a) exactly $5$ times
b) at least once.

Answer 1

(a) This question can be solved by the properties of combination,

i.e., $C_r^n{p}^k{q}^{n-k},$ where $n=8,k=5,p=q=\frac{1}{2}$ (as per given in question)

$=C_5^8{\left(\frac{1}{2}\right)}^5{\left(\frac{1}{2}\right)}^3$

$=56\ast {\left(\frac{1}{2}\right)}^8$

$=\frac{56}{256}$ (Answer)

(b) Probabilty of getting at least one head as same as the probability of not all being tails.

Thus, it is the probability of getting tails in all tosses.

Probability of getting tails in all tosses =${\left(\frac{1}{2}\right)}^8$

Probability of getting atleast 1 head = $1-{\left(\frac{1}{2}\right)}^8=\frac{255}{256}$ (Answer)