Section 8.1 - Binomial Distributions

For a situation to be considered a binomial setting, it must satisfy the following conditions:

1) Experiment is repeated a fixed number of trials and each trial is independent of the others

2) There are only two possible outcomes: success (S) and failure (F).

3) The probability of success, P(S), is the same for each trial

4) The random variable, x, counts the number of successful trials

Symbols and Notations for Binomial Settings

n = number of trials in the sample

p = probability of success in a single trial

x = count of the number of successes in n trials

this is called a binomial random variable

A binomial experiment can be symbolized as B(n,p)

x px

0

1

2

3

The probability distribution of the successes is referred to as a binomial distribution

Are these binomial experiments?1) If both parents carry the genes for the O and A blood types,

each child has a probability of 0.25 of getting two O genes and therefore having blood type O. 5 children of these parents are chosen to observe their blood type. Success is considered having blood type O.

2) Deal 10 cards from a shuffled deck and count the number, x, of red cards. Success is considered as getting a red card.

3) An engineer chooses a SRS of 10 switches from a shipment of 10,000 switches. Suppose that (unknown to the engineer) 10% of the switches in the shipment are bad. The engineer counts the number, x, of bad switches (success).

Choosing a SRS of size n from a population, where the population is much larger than the sample, the count of X

successes in the sample is approximately B(n,p)

There are several ways to find the probability of exactly k successes in n trials. One way is by the Binomial Probability Formula.

P(k) = nCx px (1 – p)n – x

This can also be done on the calculator using the

binompdf (n, p, x) Using the engineer looking for defective switches example which can be approximated as B(10,0.1) …

What is the probability that no more than 1 switch is defective?

X P(x)0

1

2

3

4

5

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10

Example: Sample surveys show that fewer people enjoy shopping than in the past. A survey asked a nationwide random sample of 2500 adults if they agreed or disagreed with the statement “ I like buying new clothes, but shopping is often frustrating and time-consuming.” The population that the poll wants to draw conclusions about is all US residents aged 18 and over. Suppose that in fact 60% of all adults US residents would say they “agree” with the statement. What is the probability that 1520 or more of the sample would agree?

If there is a large number of possible outcomes, making a table of the probability distribution is difficult. This is where the binomcdf function is useful.

Binomcdf (n, p, x) x is the upper limit of the lower tail