Final answer:
The probability of a contestant winning the game by correctly answering 4 out of 5 questions is approximately 0.39551, which is closest to answer choice (b) 0.410 after rounding.
Step-by-step explanation:
To calculate the probability of a contestant winning the game by correctly answering 4 out of 5 questions, we must consider the different ways in which this can happen. Since there is only one correct answer out of four options for each question, the probability of correctly answering a question is 1/4 or 0.25, and the probability of answering it incorrectly is 3/4 or 0.75.
Let's calculate the probability of exactly 4 correct answers out of 5 questions:
- P(exactly 4 correct answers) = (Number of ways to choose 4 questions out of 5) x (Probability of 4 correct answers and 1 incorrect answer).
The number of ways to choose 4 questions out of 5 is given by the combination formula C(5,4) = 5. The probability of getting 4 correct is (0.25)^4, and the probability of getting 1 incorrect is (0.75). Therefore:
- P(exactly 4 correct answers) = 5 x (0.25^4) x (0.75).
Now, let's calculate the probability of getting all 5 correct answers:
- P(all 5 correct answers) = (Probability of 5 correct answers) = (0.25)^5.
The contestant wins if they get exactly 4 or all 5 questions correct. So we'll add these probabilities:
- P(winning) = P(exactly 4 correct answers) + P(all 5 correct answers).
When we perform the calculation, we get:
- P(winning) = 5 x (0.25^4) x (0.75) + (0.25^5) = 0.39551 (approx).
However, none of the provided answer choices (a) 0.205, (b) 0.410, (c) 0.615, (d) 0.820 match the calculated probability. The closest answer by rounding would be (b) 0.410.