Question

1. Determine if the following describes a binomial experiment. If not, give a reason why not:

Five cards are randomly selected with replacement from a standard deck of playing cards, and the number of aces is recorded.

2. Determine if the following describes a binomial experiment. If not, give a reason why not:

Two cards are randomly selected without replacement from a standard deck of playing cards, and the number of kings (K) is recorded.

3. Determine if the following describes a binomial experiment. If not, give a reason why not:
Toss a 6-sided die until a "2" is observed.

4. Determine if a Poisson experiment is described, and select the best answer:

Suppose we knew that the average number of typos in our statistics text was 0.08 per page. The author knows that he is much more likely to make a typo on a page that has many mathematical symbols or formulas compared to pages that contain only plain text. He would like to know the probability that a randomly selected page that contains only text will contain no typos.

Answer 1

Binomial experiment is observing the number of success in n repetition of a Bernoulli trial.

1. Since we are taking card with replacement, the probability of getting ace will remain same in all five selections.

So let p be the probability of getting an ace in one draw,denote it by success. Then p=4/52=1/13.

Let X denote the number of aces in 5 selected cards ( with replacement).

Then the given experiment is binomial experiment and X has binomial distribution with

2. Since we are selecting cards without replacement, the probability of getting king (K) will change from trial to trial. Hence the experiment is not a Binomial experiment.

3. Denote occurance of side 2 as success. Let X denote the number of tosses required to get first success

( ie side 2.). Then X can take values 1,2,3,.... So X is not a finite random variable. Hence the experiment is not Binomial experiment

You may note that here the random variable is number of trials required for first success, which is a Geometric random variable.

4. Poisson distribution is used to model rare events.

Here it is given that the average number of typos in a text is 0.08 per page which is small or in oher words typo is a rare event

So we can consider the typo as a rare event. Let X be the number of typos in a randomly selected page. Then X can take values 0,1,2,3,.... and given average typos per page is 0.08 ie .

Since X can take values 0,1,2,3,.... and sice typo is a rare event, the experiement can be considered as a Poisson experiment.