Binomial Distribution

The two assumptions for a Binomial Distribution to exist are

• (a) Fixed number of trials

  (b) Probability of success is a constant in each trial

• (a) 2 mutually exclusive outcomes: success or failure

  (b) Each trial is independent of any other trial

• (a) Probability of success is a constant in each trial

  (b) Probability that each trial is independent of any other trial

• (a) Probability of success is a constant in each trial

  (b) Each trial is independent of any other trial

Answer explanation

(1) Fixed number of trials and (2) 2 mutually exclusive outcomes: success or failure are conditions and NOT assumptions for a Binomial Distribution.

Probability that each trial is independent of any other trial is incorrect! Probability is a quantity and probability CANNOT be independent of any other trials.

Correct answer is hence:
(1) Probability of success is a constant in each trial,

(2) Each trial is independent of any other trial.

Tags

CCSS.HSS.MD.A.3

CCSS.HSS.MD.A.4

For X ~ B (5, 0.3), the expectation of X is

• 5(0.3)(0.7)

• 5(0.3)

• 5(0.7)

• (0.3)^2

Answer explanation

For X ~ B (5, 0.3),

The value 5 represents n, the number of trials.

The value 0.3 represents the probability of success for the event.

To calculate the mean of X, we use np, hence the answer is 5(0.3).

Tags

CCSS.HSS.MD.A.2

For X ~ B (5, 0.3), the standard deviation of X is

• sqrt [ (5)(0.3)(0.7) ]

• (0.3)^2

• (5)(0.3)(0.7)

• 0.3

Answer explanation

For X ~ B (5, 0.3),

The value 5 represents n, the number of trials.

The value 0.3 represents the probability of success for the event.

To calculate the variance of X, we use the formula, Var (X) = np(1-p).

Hence the standard deviation of X

= sqrt (Var (X))
= sqrt [ np(1-p) ]

= sqrt [5(0.3)(0.7)]

The Australian Mathematics Competition is a multiple-choice test consisting of 25 questions. Each question has five possible answers of which only one is correct. A particular candidate decides to select his answers at random. Find the probability that he obtains 6 correct answers.

• 0.837

• 0.780

• 0.220

• 0.163

Answer explanation

For this question, X ~ B(25, 1/5).

Question is asking for P(X = 6).

Hence we use Binompdf (25, 1/5, 6).

The Australian Mathematics Competition is a multiple-choice test consisting of 25 questions. Each question has five possible answers of which only one is correct. A particular candidate decides to select his answers at random. Find the probability that he obtains less than 10 correct answers.

• 0.994

• 0.983

• 0.0173

• 0.00555

Answer explanation

For this question, X ~ B(25, 1/5).

Question is asking for P(X < 10) .

To use Binomcdf, we need to ensure the probability is of the form P(X <= a).

Thus for this question, we need to express P(X < 10) as P(X <= 9) before using Binomcdf.

The Australian Mathematics Competition is a multiple-choice test consisting of 25 questions. Each question has five possible answers of which only one is correct. A particular candidate decides to select his answers at random. Find the probability that he obtains more than 5 correct answers.

• 0.617

• 0.780

• 0.383

• 0.220

Answer explanation

For this question, X ~ B(25, 1/5).

Question is asking for P(X > 5).

We cannot apply Binomcdf as it is not of the form

P(X <= a).

We need to convert P(X > 5) = 1 - P(X <=5) first before applying Binomcdf.

A large number of light bulbs are placed in a box. 5% of the light bulbs in the box are faulty. What is the largest sample size that can be taken from the box if it is required that the probability that there are no faulty bulbs in the sample is lesser than 0.3?

 

Answer explanation

Let X ~ B (n, 0.05).

Since n is unknown and P (X = 0) < 0.3, we use table method to solve.

In GC: key in Y= Binompdf (X, 0.05, 0) and go to table. Scroll down the table to look for the first value of X that will result in the probability < 0.3.

Tags

CCSS.HSS.MD.B.6

CCSS.HSS.MD.B.7

Of the teenagers in a certain city with a large population, 35% are known to be shortsighted. If 12 random teenagers are waiting to see the doctor at a polyclinic, what is the most likely number of them being shortsighted?

Answer explanation

X ~ B (12, 0.35).

Since the question is asking for the mode, it means we need to find the value of X that gives us the highest probability.

To do that, we will use the table method to construct the probability at different values of X and compare to see which value of X gives us the highest probability.

Go to Y= and key in Binompdf (12, 0.35, X), then go to table to pick the value of X that corresponds to the highest probability.