Question

The probability of happening an event $A$ is one trial is $0.4$. The probability that the event $A$ happens at least once in three independent trials is

• A. $0.936$
• B. $0.784$
• C. $0.904$
• D. $0.216$

Answer 1

The correct option is B

$0.784$

Explanation for the correct option:

Find the probability.

Given expression,

The probability of happening of an event $A$in a trail is $P\left(A\right)$

$\therefore P\left(A\right)=0.4$

Again, if the probability of not happening of an event $A^{\text{'}}$ in a trail is $P\left(A^{\text{'}}\right)$

$$\begin{gathered} \therefore P\left(A^{\text{'}}\right)=1-0.4 \\ \Rightarrow P\left(A^{\text{'}}\right)=0.6 \end{gathered}$$ $[$$\because$$P\left(A\right)+P\left(A^{\text{'}}\right)=1]$

Now, suppose we represent $Y$ as the event ,that event $A$is not in all trails and probability is $P\left(Y\right)$

$$\begin{gathered} \therefore P\left(Y\right)=0.6\times 0.6\times 0.6 \\ \Rightarrow P\left(Y\right)={\left(0.6\right)}^3 \end{gathered}$$

Again ,if we represent $Z$ as the event that represent the happening of a event $A$ in three trails atleast one time and probability is $P\left(Z\right)$

$$\begin{gathered} P\left(Z\right)=1-P\left(Y\right) \\ \Rightarrow P\left(Z\right)=1-{\left(0.6\right)}^3 \\ \Rightarrow P\left(Z\right)=0.784 \end{gathered}$$ $[\because P\left(Y\right)+P\left(Z\right)=1]$

Therefore, The probability that the event A happens at least once in three independent trials is $0.784$