Binomial Distribution (Optional) Quiz

PDF page 272: defined as a success, while the other outcome is defined as a failure. The letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial. p + q = 1.

PDF page 272: defined as a success, while the other outcome is defined as a failure. The letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial. p + q = 1.

PDF page 272: This is a binomial experiment since it meets all three characteristics. The number of trials n = 1. There are only two outcomes, a head or a tail, of each trial. We can define a head as a success if we are measuring number of heads. For a fair coin, the probabilities of getting head or tail are both .5. So, p = q − .5. Both p and q remain the same from trial to trial. This experiment is also called a Bernoulli trial, named after Jacob Bernoulli who, in the late 1600s, studied such trials extensively. Any experiment that has characteristics two and three and where n = 1 is called a Bernoulli trial. A binomial experiment takes place when the number of successes is counted in one or more Bernoulli trials.

PDF page 272: This is a binomial experiment since it meets all three characteristics. The number of trials n = 1. There are only two outcomes, a head or a tail, of each trial. We can define a head as a success if we are measuring number of heads. For a fair coin, the probabilities of getting head or tail are both .5. So, p = q − .5. Both p and q remain the same from trial to trial. This experiment is also called a Bernoulli trial, named after Jacob Bernoulli who, in the late 1600s, studied such trials extensively. Any experiment that has characteristics two and three and where n = 1 is called a Bernoulli trial. A binomial experiment takes place when the number of successes is counted in one or more Bernoulli trials.

PDF page 273: 1 guess (you have no clue at all), the probability of guessing correct should be because there are four options and 4 1 1 3

PDF page 273: only one option is correct. So, and p = and q = 1 − p = 1 − = . Both p and q remain the same from trial to

PDF page 273: This is not a binomial experiment since the third characteristic is not met. The number of trials n = 2. There are only two outcomes, a red ball or a blue ball, of each trial. If we define selecting a red ball as a success, then selecting a blue 5 ball is a failure. The probability of getting the first ball red is since there are five red balls out of 10 balls. So, 10 5 5 5

PDF page 273: p = and q = 1 − p = 1 − = . However, p and q do not remain the same for the second trial. If the first

PDF page 273: This is not a binomial experiment since the first characteristic is not met. The number of trials n is not fixed. n could be 1 if a head appears from the first toss. n could be 2 if the first toss is a tail and the second toss is a head. So on and so forth.

PDF page 273: More examples of binomial and non-binomial experiments will be discussed in this section later. The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials.

PDF page 273: μ = np, σ = npq, σ = npq.

PDF page 273: μ = np, σ = npq, σ = npq.

PDF page 277: is the mean, μ = np = (20)(.41) = 8.2.

PDF page 277: The formula for the variance is σ = npq. The standard deviation is σ = npq .

PDF page 277: σ = (20)(.41)(.59) = 2.20.