Suppose you have 3 jars with the following contents. Jar 1 has 1 white ball and 2 black balls. Jar 2 has 2 white balls and 3 black balls. Jar 3 has 1 white ball and 4 black balls. One jar is to be selected, and then 1 ball is to be drawn from the selected jar. The probabilities of selecting the first, second, and third jars are $1 / 3,1 / 2$, and $1 / 6$ respectively. Find the probability the ball was drawn from Jar 1 , given that the ball is white. What is the probability the ball was drawn from Jar 1 , given that the ball is white? (Simplify your answer. Type an integer or a fraction.)

Step 1 ：First, we need to calculate the probability of drawing a white ball from each jar. For Jar 1, there is 1 white ball out of a total of 3 balls, so the probability is \(\frac{1}{3}\). For Jar 2, there are 2 white balls out of a total of 5 balls, so the probability is \(\frac{2}{5}\). For Jar 3, there is 1 white ball out of a total of 5 balls, so the probability is \(\frac{1}{5}\).

Step 2 ：Next, we need to calculate the total probability of drawing a white ball. This is the sum of the probabilities of drawing a white ball from each jar, each multiplied by the probability of selecting that jar. So the total probability is \(\frac{1}{3} \times \frac{1}{3} + \frac{1}{2} \times \frac{2}{5} + \frac{1}{6} \times \frac{1}{5} = \frac{1}{9} + \frac{1}{5} + \frac{1}{30} = \frac{11}{45}\).

Step 3 ：Finally, we need to calculate the probability that the ball was drawn from Jar 1 given that the ball is white. This is the probability of drawing a white ball from Jar 1, multiplied by the probability of selecting Jar 1, divided by the total probability of drawing a white ball. So the required probability is \(\frac{\frac{1}{3} \times \frac{1}{3}}{\frac{11}{45}} = \frac{1}{9} \times \frac{45}{11} = \frac{5}{11}\).

Step 4 ：So the probability the ball was drawn from Jar 1, given that the ball is white, is \(\boxed{\frac{5}{11}}\).