To get an A grade on the test, you need a total score of more than 16 points. One of the students knows the correct answer to 6 of the 20 questions. The rest she guesses at random by tossing a coin (one toss per question, as in 4B). What is the chance that she gets an A grade on the test?
To find the chance of getting an A grade on the test, we need to calculate the probability of scoring more than 16 points out of 20 questions.
Let's break down the calculation step by step:
1. The student knows the correct answer to 6 questions. This means she is guaranteed to get those questions correct, so they automatically count towards her score.
2. The remaining questions, she guesses at random by tossing a coin. Since each question has two possible outcomes (correct or incorrect), she has a 50% chance of getting each question correct by guessing.
3. The number of questions she guesses correctly follows a binomial distribution with parameters n (the number of trials) and p (the probability of success). In this case, n = 20 - 6 = 14 (the remaining questions) and p = 0.5 (the probability of guessing correctly).
4. We need to calculate the probability of scoring more than 16 points, which means getting at least 17, 18, 19, or 20 questions correct.
To calculate these probabilities, we can use the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where P(X=k) is the probability of getting exactly k questions correct, (n choose k) is the binomial coefficient, p^k is the probability of success raised to the power of k, and (1-p)^(n-k) is the probability of failure raised to the power of (n-k).
Let's calculate the probabilities for each case and then sum them up:
P(X>=17) = P(X=17) + P(X=18) + P(X=19) + P(X=20)
= [(14 choose 17) * (0.5^17) * (0.5^(14-17))] + [(14 choose 18) * (0.5^18) * (0.5^(14-18))] + [(14 choose 19) * (0.5^19) * (0.5^(14-19))] + [(14 choose 20) * (0.5^20) * (0.5^(14-20))]
= (14C17 * 0.5^17 * 0.5^-3) + (14C18 * 0.5^18 * 0.5^-4) + (14C19 * 0.5^19 * 0.5^-5) + (14C20 * 0.5^20 * 0.5^-6)
You can calculate each term in the above equation using combinations (14Ck) and the binomial probability formula. Finally, sum up the four values to get the overall probability.