1. **Stating the problem:** We want to find the probability that a participant wins the game. 2. **Understanding the game stages:** - Stage 1: Participant picks a paper from 50 papers (10 discontinuous, 40 continuous). - To proceed, the paper must be continuous. - Stage 2: Participant has 3 chances to pop pimples from 25 pieces (15 zeros, 10 ones). - The participant loses if they pop two zeros (0) in these 3 chances. - To win, the participant must pop at least two ones (1). 3. **Step 1: Probability to proceed to stage 2** $$P(\text{proceed}) = \frac{40}{50} = \frac{4}{5} = 0.8$$ 4. **Step 2: Probability to win in stage 2** We model popping pimples as drawing 3 times without replacement from 25 pieces (15 zeros, 10 ones). - Let $X$ = number of ones popped in 3 draws. - Participant wins if $X \geq 2$. 5. **Calculate probabilities for $X=2$ and $X=3$:** - Total ways to choose 3 pimples: $$\binom{25}{3} = 2300$$ - Ways to choose exactly 2 ones and 1 zero: $$\binom{10}{2} \times \binom{15}{1} = 45 \times 15 = 675$$ - Ways to choose exactly 3 ones: $$\binom{10}{3} = 120$$ 6. **Calculate probabilities:** $$P(X=2) = \frac{675}{2300} \approx 0.2935$$ $$P(X=3) = \frac{120}{2300} \approx 0.0522$$ 7. **Total probability to win stage 2:** $$P(\text{win stage 2}) = P(X=2) + P(X=3) = 0.2935 + 0.0522 = 0.3457$$ 8. **Final probability to win the game:** $$P(\text{win game}) = P(\text{proceed}) \times P(\text{win stage 2}) = 0.8 \times 0.3457 = 0.2766$$ **Answer:** The probability that the participant wins the game is approximately **0.277** (or 27.7%).