Question

What is the probability of getting at least $3$ heads?

Answer 1

Step 1: Finding the total number of outcomes

We can calculate the probability of getting at least $3$ heads by considering the total number of possibilities of getting $3$ or more than $3$ heads. As we all know, at least $3$ refers to numbers that begin with the number $3$.

When $n$ number of coins are tossed together then a total number of possible outcomes will be $2^n$.

Let us consider the case of tossing four coins. Then the total outcomes, $n\left(S\right)=2^4=16$

Step 2: Finding the probability of getting at least $3$ heads

The number of outcomes to get $r$ heads out of $n$ is ${}^{n}C_r=\frac{n!}{r!\left(n-r\right)!}$

When at least getting three heads, the outcomes are ,$4_{C_3}=4$

When at least getting four heads, the outcomes are ,${4_C}_4=1$

The number of possibility of getting at least $3$ heads means $3$ or $4$ heads,$n\left(R\right)=4+1=5$

The formula of probability is $\mathrm{Probability}=\frac{\mathrm{The}\mathrm{number}\mathrm{of}\mathrm{favorable}\mathrm{outcoms}}{\mathrm{The}\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{outcomes}}$

Then Probability of getting at least $3$ heads, $P\left(H\right)=\frac{n\left(R\right)}{n\left(S\right)}=\frac{5}{16}$

Hence,the probability of getting at least $3$ heads$=\frac{5}{16}$.