To determine the probability that Jason will win all 3 prizes, we can analyze the problem step by step.
- Total Tickets: There are 30 raffle tickets in total.
- Tickets Bought by Jason: Jason bought 10 tickets.
When calculating probabilities, we need to consider that once a ticket wins a prize, it is not returned to the draw.
Step 1: Probability of Winning the First Prize
The probability that Jason wins the first prize is the ratio of his tickets to the total tickets: \[ P(\text{Jason wins 1st}) = \frac{10}{30} = \frac{1}{3} \]
Step 2: Probability of Winning the Second Prize
If Jason wins the first prize, there are now 29 tickets left (30 - 1), and Jason still has 9 winning tickets (10 - 1): \[ P(\text{Jason wins 2nd}) = \frac{9}{29} \]
Step 3: Probability of Winning the Third Prize
If Jason wins the second prize, there are now 28 tickets left (29 - 1), and Jason still has 8 winning tickets (9 - 1): \[ P(\text{Jason wins 3rd}) = \frac{8}{28} = \frac{2}{7} \]
Step 4: Overall Probability
To find the total probability that Jason wins all three prizes, we multiply the individual probabilities: \[ P(\text{All Wins}) = P(\text{Jason wins 1st}) \times P(\text{Jason wins 2nd}) \times P(\text{Jason wins 3rd}) \]
Calculating this: \[ P(\text{All Wins}) = \left(\frac{1}{3}\right) \times \left(\frac{9}{29}\right) \times \left(\frac{2}{7}\right) \]
Now simplify it:
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Multiply the numerators: \[ 1 \times 9 \times 2 = 18 \]
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Multiply the denominators: \[ 3 \times 29 \times 7 = 609 \]
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So, the probability that Jason wins all 3 prizes is: \[ P(\text{All Wins}) = \frac{18}{609} \]
Step 5: Simplifying the Fraction
Both the numerator and the denominator can be divided by 3: \[ \frac{18 \div 3}{609 \div 3} = \frac{6}{203} \]
Thus, the probability that Jason will win all 3 prizes is: \[ \boxed{\frac{6}{203}} \]