Question

Suppose that Ann selects a ball by first picking one of two boxes at random and then selecting a ball from this box. The first box contains three orange balls and four black balls, and the second box contains five orange balls and six black balls. What is the probability that Ann picked a ball from the second box if she has selected an orange ball?

Short answer

Answer

The probability that Ann picked a ball from the second box if she has selected an orange ball is $\frac{35}{68}$

Step-by-step solution

Step-1 Given Information

1. The first box contains three orange balls and four black balls
2. The second box contains five orange balls and six black balls.

Step-2 Definition and Formula

E= {an orange ball is chosen}.

F= {a ball is picked from the second box}.

The formula use to find the probability that Ann picked a ball from the second box if she has selected an orange ball is -

$\mathit{P}\mathbf{\left(}\mathit{F}\mathbf{\right|}\mathit{E}\mathbf{)}\mathbf{=}\frac{\mathbf{P}\mathbf{\left(}\mathbf{E}\mathbf{\right|}\mathbf{F}\mathbf{\left)}\mathbf{P}\mathbf{\right(}\mathbf{F}\mathbf{)}}{\mathbf{P}\mathbf{\left(}\mathbf{E}\mathbf{\right|}\mathbf{F}\mathbf{\left)}\mathbf{P}\mathbf{\right(}\mathbf{F}\mathbf{)}\mathbf{+}\mathbf{P}\mathbf{(}\mathbf{E}\mathbf{\left|}\overline{\mathbf{F}}\mathbf{\right)}\mathbf{P}\mathbf{\left(}\overline{\mathbf{F}}\mathbf{\right)}}$

Step-3 Find the P(E|F) and P(E|F¯) 

E= {an orange ball is chosen}.

F= {a ball is picked from the second box}.

$$\begin{gathered} P\left(E\right)=P\left(\overline{E}\right) \\ =\frac{1}{2} \end{gathered}$$

Now,

$$\begin{gathered} P\left(E\right|F)=\frac{5}{(5+6)} \\ =\frac{5}{11} \\ P(E\left|\overline{F}\right)=\frac{3}{(3+4)} \\ ==\frac{3}{7} \end{gathered}$$

Step-4 Find the  P(F|E)

So,

$$\begin{gathered} P\left(F\right|E)=\frac{P\left(E\right|F\left)P\right(F)}{P\left(E\right|F\left)P\right(F)+P(E\left|\overline{F}\right)P\left(\overline{F}\right)} \\ =\frac{\left(\frac{5}{11}\right)\left(\frac{1}{2}\right)}{\left(\frac{5}{11}\right)\left(\frac{1}{2}\right)+\left(\frac{3}{7}\right)\left(\frac{1}{2}\right)} \\ =\frac{35}{68} \end{gathered}$$

Hence,

The probability that Ann picked a ball from the second box if she has selected an orange ball is $\frac{35}{68}$