Question

A fair coin is tossed repeatedly. The probability of getting a result in the fifth toss different from those obtained in the first four tosses is

• A. $\frac{1}{2}$
• B. $\frac{1}{32}$
• C. $\frac{31}{32}$
• D. $\frac{1}{16}$

Answer 1

The correct option is D $\frac{1}{16}$

Let us find the probability of Heads appearing in the toss since $P\left(H\right)=P\left(T\right)=\frac{1}{2}$ for an unbiased coin, it will not affect or introduce any error in the result.
Thus
If a coin is tossed Once.
$P\left(H\right)=\frac{1}{2}$.
If a coin is tossed twice.
The probability of head appearing twice is
$=\frac{1}{2}.\frac{1}{2}=\frac{1}{4}$
If a coin is tossed thrice.
The probability of head appearing thrice is
$=\frac{1}{2}.\frac{1}{2}.\frac{1}{2}=\frac{1}{8}$
If a coin is tossed 4 times.
The probability of head appearing 4 times is
$=\frac{1}{2}.\frac{1}{2}.\frac{1}{2}.\frac{1}{2}=\frac{1}{16}$
Therefore we can generalize
If coin is tossed n times.
The probability of head appearing n times will
$=\frac{1}{2}.\frac{1}{2}....=\frac{1}{2^n}$