# Chapter 6 1 2 3 4 5 6 7 8 9 10 11 12 13 ## Objectives

By the end of this moral, you will be able to …

1. determine whether a probability experiment is a binomial experiment
2. compute probabilities of binomial experiments
3. compute and interpret the mean and standard deviation of a binomial
random variable

For a promptly overview of this segment, watch this short video recording summary :

## Binomial Experiments

In the survive section, we talked about some specific examples of random variables. In this adjacent section, we deal with a particular type of random variable called a binomial random variable. Random variables of this type have several characteristics, but the key one is that the experiment that is being performed has lone two possible outcomes – success or failure .
An case might be a dislodge bang in soccer – either the player scores a goal or she does n’t. Another exemplar would be a flip mint – it ‘s either heads or tails. A multiple choice test where you ‘re wholly guessing would be another exercise – each interview is either right or wrong .
Let ‘s be specific about the other key characteristics as well :

#### Criteria for a Binomial Probability Experiment

A binomial experiment is an experiment which satisfies these four conditions :

• A fixed number of trials
• Each trial is independent of the others
• There are only two outcomes
• The probability of each outcome remains constant from trial to trial.

In short circuit : An experiment with a sterilize total of independent trials, each of which can only have two possible outcomes .
( Since the trials are independent, the probability remains constant. )
If an experiment is a binomial experiment, then the random variable star X = the number of successes is called a binomial random variable .
Let ‘s look at a couple examples to check your understand .

example 1 Source: reference : stock.xchng Consider the experiment where three marbles are drawn without successor from a bag containing 20 crimson and 40 blue marbles, and the count of red marbles drawn is recorded. Is this a binomial experiment ?
No ! The key here is the lack of independence – since the marbles are drawn without substitution, the marble withdraw on the first will affect the probability of former marbles .

model 2
A fair six-sided die is rolled ten times, and the number of 6 ‘s is recorded. Is this a binomial experiment ?
Yes ! There are fix count of trials ( ten rolls ), each roll is freelancer of the others, there are alone two outcomes ( either it ‘s a 6 or it is n’t ), and the probability of rolling a 6 is changeless .

## The Binomial Distribution

once we determine that a random variable star is a binomial random variable, the adjacent question we might have would be how to calculate probabilities .
Let ‘s consider the experiment where we take a multiple-choice quiz of four questions with four choices each, and the subject is something we have absolutely no cognition. Say … theoretical astrophysics. If we let X = the number of decline answer, then X is a binomial random variable because

• there are a fixed number of questions (4)
• the questions are independent, since we’re just guessing
• each question has two outcomes – we’re right or wrong
• the probability of being correct is constant, since we’re guessing: 1/4

So how can we find probabilities ? Let ‘s look at a tree diagram of the situation : Finding the probability distribution of X involves a couple key concepts. First, notice that there are multiple ways to get 1, 2, or 3 questions correct. In fact, we can use combinations to figure out how many ways there are ! Since P ( X=3 ) is the same regardless of which 3 we get decline, we can precisely multiply the probability of one course by 4, since there are 4 ways to get 3 correct .
not only that, since the questions are independent, we can merely multiply the probability of getting each one discipline or incorrect, so P ( ) = ( 3/4 ) 3 ( 1/4 ). Using that concept to find all the probabilities, we get the follow distribution :

 x P(x) 0 1 2 3 4 We should notice a couple very crucial concepts. First, the number of possibilities for each measure of X gets multiplied by the probability, and in general there are 4Cx ways to get X correct. second, the exponents on the probabilities represent the number chastise or wrong, so do n’t stress out about the recipe we ‘re about to show. It ‘s basically :
P ( X ) = ( ways to get X successes ) • ( prob of success ) successes• ( prob of bankruptcy ) failures

#### The Binomial Probability Distribution Function

The probability of obtaining ten successes in newton independent trials of a binomial experiment, where the probability of success is p, is given by Where x = 0, 1, 2, …, n

## Technology

here ‘s a quick overview of the formula for finding binomial probabilities in StatCrunch .

 Click on Stat > Calculators > Binomial Enter newton, phosphorus, the allow equality/inequality, and x. The number below shows P ( X≥3 ) if n=4 and p=0.25. Let ‘s try some examples .

example 3
Consider the example again with four multiple-choice questions of which you have no cognition. What is the probability of getting precisely 3 questions correct ?
For this exercise, n=4 and p=0.25. We want P ( X=3 ) .
We can either use the defining recipe or software. The picture below shows the calculation using StatCrunch . So it looks like P ( X=3 ) ≈ 0.0469
( We normally round to 4 decimal places, if necessary. )

exercise 4 Source: source : stock.xchng A basketball actor traditionally makes 85 % of her free throws. Suppose she shoots 10 baskets and counts the number she makes. What is the probability that she makes less than 8 baskets ?

If X = the count of made baskets, it ‘s reasonable to say the distribution is binomial. ( One could make an argument against independence, but we ‘ll assume our player is n’t affected by former makes or misses. )
In this example, n=10 and p=0.85. We want P ( X < 8 ) . P ( X < 8 ) = P ( X≤7 ) = P ( X=0 ) + P ( X=1 ) + ... + P ( X=7 ) quite than computing each one independently, we 'll use the binomial calculator in StatCrunch . It looks like the probability of making less than 8 baskets is about 0.1798 .

model 5
traditionally, about 70 % of students in a particular Statistics course at ECC are successful. Suppose 20 students are selected at random from all previous students in this run. What is the probability that more than 15 of them will have been successful in the course ?
Let ‘s do a quick overview of the criteria for a binomial experiment to see if this fits .

• A fixed number of trials – The students are our trials.
• Each trial is independent of the others – Since they’re randomly selected, we can assume they are independent of each other.
• There are only two outcomes – Each student either was successful or was not successful.
• The probability of each outcome remains constant from trial to trial. – Because the students were independent, we can assume this probability is constant.

If we let X = the number of students who were successful, it does look like X follows the binomial distribution. For this example, n=20 and p=0.70 .
Let ‘s use StatCrunch for this calculation : thus P ( more than 15 were successful ) ≈ 0.2375 .

## The Mean and Standard Deviation of a Binomial Random Variable

Let ‘s consider the basketball actor again. If she takes 100 free throws, how many would we expect her to make ? ( Remember that she historically makes 85 % of her complimentary throws. )
The answer, of course, is 85. That ‘s 85 % of 100.

We could do the lapp with any binomial random variable. In Example 5, we said that 70 % of students are successful in the Statistics naturally. If we randomly sample 50 students, how many would we expect to have been successful ?
again, it ‘s fairly square – 70 % of 50 is 35, so we ‘d expect 35 .