Question

Six Coins Are Tossed Simultaneously. What Is The Probability Of Getting Three Heads?$p$

Answer 1

Let p represents the probability of getting head in a toss of a coin.

So $p=\frac{1}{2}$

Let x denote the random variable representing the number of heads in $6$ tosses of a coin.

Probability of getting $r$ sixes in $n$ tosses of a fair is given by,

$p\left(x=r\right){=}^6{C}_r{\left(\frac{1}{2}\right)}^r{\left(\frac{1}{2}\right)}^{6-r}$

Probability of getting three heads :

$$\begin{gathered} p\left(x=3\right){=}^6{C}_3{\left(\frac{1}{2}\right)}^3{\left(\frac{1}{2}\right)}^{6-3} \\ =\frac{6!}{3!\left(6-3\right)!}{\left(\frac{1}{2}\right)}^3{\left(\frac{1}{2}\right)}^3 \\ =20\times {\left(\frac{1}{2}\right)}^6 \\ =20\times \frac{1}{2^6} \\ =20\times \frac{1}{64} \\ =\frac{5}{16} \end{gathered}$$

Hence, the probability of getting three heads is $\frac{5}{16}$