Question

John makes random guesses on his multiple-choice test, which has five options for each question. Let the random variable X be the number of guesses taken before guessing correctly.

Assuming the guesses are independent, find the probability that he doesn't guess correctly until his 6th guess.

0.0655

0.0789

0.3521

0.3277

Answer 1

Answer:

$P(X=6 , r=1) = (6-1 C 1-1)0.2^1 (1-0.2)^{6-1}= (5C0) 0.2^1 0.8^5 =0.0655$

So the correct option would be:

0.0655

Step-by-step explanation:

Previous concepts

A negative binomial random variable "is the number X of repeated trials to produce r successes in a negative binomial experiment. The probability distribution of a negative binomial random variable is called a negative binomial distribution, this distribution is known as the Pascal distribution".

And the probability mass function is given by:

$P(X=x) = (x-1 C r-1)p^r (1-p)^{x-r}$

Where r represent the number successes after the x trials and p is the probability of a success on any given trial.

Solution to the problem  

Let X the random variable that represent the number of guesses taken before guessing correctly.

For this case the random variable X follows a negative binomial distribution given by:

$X \sim Neg Bin(n=6, r=1, p=\frac{1}{5}=0.2)$

And we want to find the probability P(X=1) and if we replace in the mass function we got:

$P(X=6 , r=1) = (6-1 C 1-1)0.2^1 (1-0.2)^{6-1}= (5C0) 0.2^1 0.8^5 =0.0655$

So the correct option would be:

0.0655

Answer 2

The probability that he doesn't guess correctly until his 6th guess is 0.0655 and this can be determined by using the given data.

Given :

• John makes random guesses on his test. Each question has five options.
• Independent guesses.

The following steps can be used in order to determine the probability that he doesn't guess correctly until his 6th guess:

Step 1 - According to the given data, each question has five options.

Step 2 - So, the probability that he doesn't guess correctly until his 6th guess can be calculated as:

$P(X=6,r=1)= \;^{6-1}C_{1-1}\times (0.2)^1\times (1-0.2)^{6-1}$

Step 3 - Simplify the above expression.

$P(X=6,r=1)= \;^{5}C_{0}\times (0.2)\times (0.8)^{5}$

$P(X=6,r=1)= 0.0655$

So, the probability that he doesn't guess correctly until his 6th guess is 0.0655.

For more information, refer to the link given below:

https://brainly.com/question/743546