Question

Lucky penny? Harris Interactive reported that $33\%$of U.S. adults believe that

finding and picking up a penny is good luck. Assuming that responses from different

individuals are independent, what is the probability of randomly selecting $10$U.S. adults

and finding at least $1$person who believes that finding and picking up a penny is good

luck?

Short answer

Probability of randomly selecting $10$U.S. adults and finding at least $1$person who believes that finding and picking up a penny is good luck is $98.177\%$

Step-by-step solution

Given information

We have been given that $33\%$of United States' adults believe in finding and picking up a penny is a sign of good luck.

We need to find out the probability of randomly selecting $10$U.S. adults and finding at least $1$person who believes that finding and picking up a penny is good luck

Explanation

Let A$=$Adults believe that finding and picking up a penny is good luck.

Let B $=$At least $1$of $10$adults believe that finding and picking up a penny is good luck.

Let An$=$One adult believe that finding and picking up a penny is good luck.

Let Bn $=$None of the $10$adults believe that finding and picking up a penny is good luck.

$$\begin{gathered} P\left(A\right)=33\%=0.33 \\ P\left(An\right)=1-P\left(A\right)=1-0.33=0.67 \end{gathered}$$

Using the multiplication rule for independent events

$$\begin{gathered} P\left(Bn\right)=P\left(An\right)\times 10 \\ =P{\left(An\right)}^{10} \\ =0.{67}^{10} \\ \approx 0.01823 \end{gathered}$$

Using complement rule

$$\begin{gathered} P\left(B\right)=1-P\left(Bn\right) \\ =1-0.01823 \\ =0.98177 \\ =98.177\% \end{gathered}$$