Question

A really tough trivia quiz has 50 multiple-choice questions, each with four possibleanswers. Use a normal approximation to calculate the probability of randomly guessing the correct answers for at least half of the questions.

Answer 1

To calculate the probability of guessing the correct answers for at least half of the questions, use the normal approximation. Find the mean and standard deviation of the binomial distribution, then calculate the z-score. Use a standard normal distribution table or calculator to find the probability associated with the z-score.

Step-by-step explanation:

To calculate the probability of randomly guessing the correct answers for at least half of the questions, we can use the normal approximation. In this case, we have 50 multiple-choice questions, each with four possible answers. The probability of guessing the correct answer for each question is 1/4 = 0.25.

To find the probability of guessing at least half correctly, we can use the binomial distribution and its respective formula. However, for large sample sizes like this, we can use the normal approximation to the binomial distribution.

First, we need to find the mean and standard deviation of the binomial distribution. The mean (μ) is given by μ = n * p, where n is the number of trials (50) and p is the probability of success (0.25). μ = 50 * 0.25 = 12.5.

The standard deviation (σ) is given by σ = sqrt(n * p * (1 - p)). σ = sqrt(50 * 0.25 * (1 - 0.25)) = sqrt(9.375) ≈ 3.06.

Next, we need to calculate the z-score for the desired probability. The z-score formula is z = (x - μ) / σ, where x is the number of correct answers we want to calculate the probability for.

In this case, we want to find the probability of guessing at least half correctly, which means a minimum of 25 correct answers (50 * 0.5). So, x = 25.

Now we can calculate the z-score: z = (25 - 12.5) / 3.06 ≈ 4.08.

Finally, we can use a standard normal distribution table or calculator to find the probability associated with the z-score. The probability of guessing at least half correctly is approximately 1 - P(z < 4.08).

Note: Make sure to round final probabilities to four decimal places.

Answer 2

0.00004

Step-by-step explanation:

Given that:

n = 50

Number of options = 4 ; 1 is the correct answer

Hence ;

p = 1/4 = 0.25

Using normal approximation :

Mean (m) = np = 50 * 0. 25 = 12.5

Standard deviation (s) = √(n*p*q)

q = 1 - p = 1 - 0. 25 = 0.75

s = √(50*0.25*0.75) = 3.062

At least half the question :

Half the question = 1/2 * 50 = 25

P(X ≥ 25) = P(X ≥ 24.5)

(x - m) / s

P(X ≥ 24.5) = (24.5 - 12.5) / 3.062

= 12 / 3.062

= 3.919

P(Z ≥ 3.919) = 0.00004 (Z probability calculator)