Let's break down the problem step by step to find the probability that a randomly selected ticket will award a larger prize.
### Step 1: Understanding the Given Information
- Total number of tickets: [tex]\(10\)[/tex]
- Number of winning tickets: [tex]\(6\)[/tex]
- Out of the winning tickets, one out of every three awards a larger prize.
### Step 2: Probability of Selecting a Winning Ticket
First, we need to find the probability that a randomly chosen ticket is a winning ticket. The probability [tex]\(\text{P(Winning Ticket)}\)[/tex] is calculated as follows:
[tex]\[
\text{P(Winning Ticket)} = \frac{\text{Number of Winning Tickets}}{\text{Total Number of Tickets}} = \frac{6}{10} = 0.6
\][/tex]
### Step 3: Probability of a Larger Prize Given a Winning Ticket
Next, we need to find the probability that a winning ticket awards a larger prize. This is given as one out of every three winning tickets:
[tex]\[
\text{P(Larger Prize | Winning Ticket)} = \frac{1}{3}
\][/tex]
### Step 4: Combined Probability
To find the overall probability that a randomly chosen ticket will award a larger prize, we multiply the probability of selecting a winning ticket by the probability that the winning ticket awards a larger prize:
[tex]\[
\text{P(Larger Prize)} = \text{P(Winning Ticket)} \times \text{P(Larger Prize | Winning Ticket)} = 0.6 \times \frac{1}{3} = 0.2
\][/tex]
### Step 5: Converting to Fraction
Now we convert the decimal probability into a simplified fraction:
[tex]\[
0.2 = \frac{2}{10} = \frac{1}{5}
\][/tex]
So, the probability that a randomly chosen ticket will award a larger prize is:
[tex]\(\boxed{\frac{1}{5}}\)[/tex]
