Question

What is the probability of getting all tails?

Answer 1

Step- 1:When a coin is tossed $2$ times:

The probability of an event is,

$\mathbf{\text{P}}\left(E\right)\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{Number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{favourable}\mathbf{}\mathbf{outcomes}}{\mathbf{Total}\mathbf{}\mathbf{number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{outcomes}}$

The total outcomes when a coin is tossed $n$ times is ${\mathbf{2}}^{\mathbf{n}}$.

When a coin is tossed $2$ times, the possible outcomes are:

$HT,HH,TH,TT$

The total number of outcomes $=4$

Let $E$ be the event of getting $2$ tails.

$E=\left\{TT\right\}$

The number of a favorable outcomes $=1$

Then, the probability of getting all tails is,

$P\left(E\right)=\frac{1}{4}$

Step- 2:When a coin is tossed $3$ times:

When a coin is tossed $3$ times, the possible outcomes are:

$HHH,HHT,HTH,THH,TTH,THT,HTT,TTT$

The total number of outcomes $=8$

Let $E$ be the event of getting $3$ tails.

$E=\left\{TTT\right\}$

The number of a favorable outcomes $=1$.

Then, the probability of getting all tails is,

$P\left(E\right)=\frac{1}{8}$

Step- 3:When a coin is tossed $n$ times:

When a coin is tossed $n$ times

The total number of outcomes $\mathbf{=}{\mathbf{2}}^{\mathbf{n}}$

Let $E$ be the event of getting all tails.

$E=\left\{TTT\cdots n\mathrm{times}\right\}$

The number of a favorable outcomes $=1$.

Then, the probability of getting all tails is,

$P\left(E\right)=\frac{1}{2^n}$

Hence, the probability of getting all tails is $\frac{1}{2^n}$ where $n$ is the number of times a coin tossed.