Question

A student takes a true-false test that has 10 questions and guesses randomly at each answer. Let X be the number of questions answered correctly. Find P(Fewer than 3). Round your answer to 2 decimal places.

Answer 1

To find the probability of the student passing the test with at least a 70 percent, we can use the binomial probability formula. The probability of the student passing the test with at least 70 percent is 0.1719 (rounded to 2 decimal places).

Step-by-step explanation:

To find the probability of the student passing the test with at least a 70 percent, we need to find the probability of the student answering 7, 8, 9, or 10 questions correctly out of the 10 questions. Since the student randomly guesses at each answer, the probability of guessing correctly is 0.5. Now we can calculate the probability using the binomial probability formula:

P(X ≥ 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)

P(X = k) = C(10, k) * (0.5)^k * (0.5)^(10-k), where C(n, r) is the binomial coefficient (n choose r).

Calculating each probability and summing them up, we get P(X ≥ 7) = 0.171875. Therefore, the probability of the student passing the test with at least 70 percent is 0.1719 (rounded to 2 decimal places).

Answer 2

P(Fewer than 3) = 0.05.

Step-by-step explanation:

We are given that a student takes a true-false test that has 10 questions and guesses randomly at each answer.

The above situation can be represented through Binomial distribution;

`P(X=r) = \binom{n}{r}p^(r) (1-p)^(n-r) ; x = 0,1,2,3,.....`

where, n = number of trials (samples) taken = 10 questions

r = number of success = fewer than 3

p = probability of success which in our question is probability

that question is answered correctly, i.e; 50%

LET X = Number of questions answered correctly

So, it means X ~ Binom(n = 10, p = 0.50)

Now, Probability that Fewer than 3 questions are answered correctly is given by = P(X < 3)

P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)

=
`\binom{10}{0}* 0.50^(0) * (1-0.50)^(10-0)+ \binom{10}{1}* 0.50^(1) * (1-0.50)^(10-1)+ \binom{10}{2}* 0.50^(2) * (1-0.50)^(10-2)`

=
`1 * 0.50^(10) + 10 * 0.50^(10) +45 * 0.50^(10)`

= 0.05

Hence, the P(Fewer than 3) is 0.05.