Question

A fair coin is tossed a fixed number of items. If the probability of getting$7$ heads is equal to that of getting $9$ heads, then the probability of getting $3$ heads is

• A. $\frac{35}{2^{12}}$
• B. $\frac{35}{2^{14}}$
• C. $\frac{7}{2^{12}}$
• D. None of these

Answer 1

The correct option is A

$\frac{35}{2^{12}}$

Explanation for correct option:

Step-1: Define events:

Number of times coin tossed $=n$

Probability of getting heads $\text{'}p\text{'}=\frac{1}{2}$

Probability of getting tails $\text{'}q\text{'}=\frac{1}{2}\mathbf{}\left[\because p+q=1\right]$

Step-2 : Calculate the value of $\text{'}n\text{'}$

As per the condition given

Probability of getting seven heads $=$Probability of getting nine heads

${}^{n}C_r{\left(\frac{1}{2}\right)}^7{\left(\frac{1}{2}\right)}^{n-7}={}^{n}C_9{\left(\frac{1}{2}\right)}^9{\left(\frac{1}{2}\right)}^{n-9}\left[\because {}^{n}C_r{p}^r{q}^{n-r}\right]$

$$\begin{gathered} \Rightarrow {}^{n}C_7{\left(\frac{1}{2}\right)}^n={}^{n}C_9{\left(\frac{1}{2}\right)}^n \\ \Rightarrow {}^{n}C_7={}^{n}C_9 \\ \Rightarrow n=7+9\mathbf{\left[}\mathbf{\because }{}^{\mathbf{n}}\mathit{C}_{\mathbf{r}}\mathbf{}\mathbf{=}{}^{\mathbf{n}}\mathit{C}_{\mathbf{r}\mathbf{+}\mathbf{1}}\mathbf{}\mathbf{\Rightarrow }\mathbf{n}\mathbf{}\mathbf{=}\mathbf{}\mathbf{r}\mathbf{+}\mathbf{r}\mathbf{+}\mathbf{1}\mathbf{\right]} \\ \Rightarrow n=16 \end{gathered}$$

Step3. Probability of getting three heads:

Substitute the value of $\mathrm{n}=16,\mathrm{r}=3$ in ${}^{\mathrm{n}}\mathrm{C}_{\mathrm{r}}{\mathrm{p}}^{\mathrm{r}}{\mathrm{q}}^{\mathrm{n}-\mathrm{r}}$

${}^{16}C_3{\left(\frac{1}{2}\right)}^3{\left(\frac{1}{2}\right)}^{16-3}$

$$\begin{gathered} ={}^{16}C_3{\left(\frac{1}{2}\right)}^{16} \\ =\frac{16!}{\left(16-3\right)!3!}{\left(\frac{1}{2}\right)}^{16}\left[\mathbf{\because }{}^{\mathbf{n}}\mathit{C}_{\mathbf{r}}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}\mathbf{r}\mathbf{!}}\right] \\ =\frac{\mathbf{35}}{{\mathbf{2}}^{\mathbf{12}}} \end{gathered}$$

Hence, option(A) is the correct answer.