5.2 The Binomial Distribution

anytime an outcome is only 2 possibilities, it may be a

binomial experirament

Rounding Rule:

round binomial probabilities to three decimal places. ≈0.167

A certain insecticide kills 60% of all insects in laboratory experiments. A sample of 15 insects is exposed to the insecticide in a particular experiment. What is the probability that exactly 8 insects will survive? Round your answer to four decimal places.

***asking for survivors - if .6 die, then .4 live n=15 p=.4 x=8 Use your table =.1181

Determine whether or not the given procedure results in a binomial distribution. If not, identify which condition is not met. Rolling a six-sided die 24 times and recording the number of odd numbers rolled.

yes it is a binomial distribution

Assume the random variable X has a binomial distribution with the given probability of obtaining a success. Find the following probability, given the number of trials and the probability of obtaining a success. P(X≤4), n=6, p=0.2

***Don't forget to include the ZERO*** 0 = <4

binomial experiment:

a random experiment which satisfies all of the following conditions: 1. There are only two outcomes on each trial of the experiment. One of the outcomes is usually referred to as a success, and the other as a failure. 2. The experiment consists of n identical trials as described earlier. 3. The probability of success on any one trial is denoted by p and does not change from trial to trial. Note that the probability of a failure is 1−p and also does not change from trial to trial. 4. The trials are independent. 5. The binomial random variable is the count of the number of successes in n trials.

Assume the random variable X has a binomial distribution with the given probability of obtaining a success. Find the following probability, given the number of trials and the probability of obtaining a success. Round your answer to four decimal places. P(X>4), n=6, p=0.3

n=6 p=.3 x=>4 [5,6] Use your table and subtract values from value of x=6 ≈.0156

Binomial distribution

number of so called successes in a number of trials experiments in which there is a fixed number of repeated trials having only two outcomes and one of the outcomes is counted For example: A simple and popular binomial experiment is to toss a coin for a fixed number of trials and count the number of heads. whatever you are looking for is the success!

The Shape of a Binomial Distribution

shape of the distribution depends upon the parameters n and p.

If p is large the distribution tends to be

skewed with a long tail on the left (i.e., there are more successes when x is larger).

If p is small the distribution tends to be

skewed with a tail on the right (i.e., there are more successes when x is smaller).

The standard deviation of a binomial random variable is

the square root of the variance. Therefore,

To find the variance of a binomial random variable

use the expression σ2=np(1−p)

When all of the binomial distribution conditions are met, the following formula can be used to determine the probability of obtaining x successes out of n trials.

Formula: Probability for the Binomial Distribution

A die is rolled 6 times and the number of times a 5 is rolled is recorded. Determine whether or not this procedure results in a binomial distribution.

Let's look at the conditions that must be met for an experiment to be a binomial experiment. 1. There are only two outcomes on each trial of the experiment. One of the outcomes is usually referred to as a success, and the other as a failure. This condition is met because you either roll a 5 or you do not roll a 5. 2. The experiment consists of n identical trials. The experiment will only be performed 6 times, with each trial being identical to each other. 3. The probability of success on any one trial is denoted by p and does not change from trial to trial. The probability of rolling a 5 is 1/6 and does not change from trial to trial. 4. The trials are independent. The outcome of one roll will not affect other rolls. 5. The binomial random variable is the count of the number of successes in n trials. The variable of interest is the count of the number of 5's in 6 rolls. Since all conditions are met, the procedure is a binomial distribution.

Binomial Distribution Function

The probability distribution of x successes can be calculated using the classical method of listing the outcomes or simple events of n trials. If a coin is tossed three times, there are eight different possibilities and hence eight simple events. In the following table, the events are grouped according to the number of heads contained in each simple event - this becomes unnecessarily tedious if the value of n is increased

The Magazine Mass Marketing Company has received 12 entries in its latest sweepstakes. They know that the probability of receiving a magazine subscription order with an entry form is 0.60. What is the probability that no more than 8 of the entry forms will include an order?

Use the table!!! n= 12 p= .6 x= ≤ 8 using the table, look up the amounts in the column for n=12 corresponding w/ p=.6 and x= 9 (.1419), 10 (.0639), 11 (.0174), & 12 (.0022). Sum all 4 values together to get .2254 then plug it into the formula: (1-.2254) = .7746

If p is near 0.5 the distribution is

symmetrical.

The number of heads is a count that may be modeled as a

binomial random variable.

Expected Value and Standard Deviation of a Binomial

by utilizing a special characteristic of the binomial distribution: the expected value, also known as the mean, of a binomial random variable can be computed using the expression E(X)=μ=np n and p are the parameters of the binomial distribution ***only valid for binomial random variables***

What is the probability of getting exactly 10 tails in 18 coin tosses?

in your calculator: press 2nd - DISTR scroll down to A: binompdf( - press enter type in # of trials (n), P: (probability), and x value (successes) press enter on Paste press enter again = .1669

Assume the random variable X has a binomial distribution with the given probability of obtaining a success. Find the following probability, given the number of trials and the probability of obtaining a success. Round your answer to four decimal places. P(X=5) n=18 p=0.3

in your calculator: press 2nd - DISTR scroll down to A: binompdf( - press enter type in # of trials (n), P: (probability), and x value (successes) press enter on Paste press enter again = .2017