Q.3.38 - A First Course in Probability

Question

Urn $A$ has $5$ white and $7$ black balls. Urn $B$ has $3$ white and $12$ black balls. We flip a fair coin. If the outcome is heads, then a ball from urn $A$ is selected, whereas if the outcome is tails, then a ball from urn $B$ is selected. Suppose that a white ball is selected. What is the probability that the coin landed tails?

Step-by-Step Solution

Answer

The probability that the coin landed tails is $\frac{12}{37}$.

Step 1: Given information

Urn $A$ has $5$ white and $7$ black balls. Urn $B$ has $3$ white and $12$ black balls. We flip a fair coin. If the outcome is heads, then a ball from urn $A$ is selected, whereas if the outcome is tails, then a ball from urn $B$ is selected.

Step 2: Solution

Urn $A$ $5$ white

$7 Black \Rightarrow P\left(white\right)=\frac{5}{12}$

Urn $B$ $3$ White

$12 Black \Rightarrow P\left(white\right)=\frac{3}{15}$

$P\left(\text{ Tail | white }\right)=\frac{\frac{1}{2}\times \frac{3}{15}}{\frac{1}{2}\times \frac{3}{15}+\frac{1}{2}\times \frac{5}{12}}$

$=\frac{3}{30}\times \frac{240}{74}$

$=\frac{12}{37}$

Step 3: Final answer

The probability that the coin landed tails is $\frac{12}{37}$.