In the given problem, we need to find the probability that a randomly chosen ticket will award a larger prize.
Let's break down the problem step-by-step:
1. Determine the probability of picking a winning ticket:
- We know that six out of every ten tickets are winning tickets.
- Thus, the probability of picking a winning ticket is:
[tex]\[
P(\text{winning ticket}) = \frac{6}{10} = 0.6
\][/tex]
2. Determine the probability of getting a larger prize given that the ticket is winning:
- We are given that one out of every three winning tickets awards a larger prize.
- Thus, the probability of getting a larger prize given that the ticket is winning is:
[tex]\[
P(\text{larger prize} \mid \text{winning ticket}) = \frac{1}{3} \approx 0.333
\][/tex]
3. Calculate the overall probability of getting a larger prize:
- We are asked to find the probability that a randomly chosen ticket will award a larger prize. This can be found using the chain rule of conditional probability:
[tex]\[
P(\text{larger prize}) = P(\text{winning ticket}) \times P(\text{larger prize} \mid \text{winning ticket})
\][/tex]
4. Substitute the values into the formula:
- Using the values from step 1 and step 2:
[tex]\[
P(\text{larger prize}) = 0.6 \times 0.333 \approx 0.2
\][/tex]
So, the probability that a randomly chosen ticket will award a larger prize is approximately [tex]\(0.2\)[/tex]. In fraction form, this is:
[tex]\[
\frac{1}{5} = 0.2
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{1}{5}}
\][/tex]
