Binomial Distribution (Optional) Quiz

PDF page 255: There are three characteristics of a binomial experiment:

PDF page 255: is 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 255: is 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 256: 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 256: 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 256: This is a binomial experiment since it meets all three characteristics. The number of trials n = 5. There are only two outcomes, head or tail, of each trial. If we define head as a success, then p = q = 0.5. Both p and q remain the same for each trial. Since n = 5, this experiment is not a Bernoulli trial although it meets the characteristics two and three.

PDF page 256: 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 ball is a failure. The probability of getting the first ball red is since there are five red balls out of 10 balls. So, and . However, p and q do not remain the same for the second trial. If the first ball selected is red, then the probability of getting the second ball red is since there are only four red balls out of nine balls. But if the first ball selected is blue, then the probability of getting the second ball red is since there are still five red balls out of nine balls.

PDF page 256: 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 256: 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. 2

PDF page 257: At ABC High School, the withdrawal rate from an elementary physics course is 30 percent for any given term. This implies that, for any given term, 70 percent of the students stay in the class for the entire term. The random variable X = the number of students who withdraw from the randomly selected elementary physics class. Since we are measuring the number of students who withdrew, a success is defined as an individual who withdrew.

PDF page 257: Suppose you play a game that you can only either win or lose. The probability that you win any game is 55 percent, and the probability that you lose is 45 percent. Each game you play is independent. If you play the game 20 times, write the function that describes the probability that you win 15 of the 20 times. Here, if you define X as the number of wins, then X takes on the values 0, 1, 2, 3, . . ., 20. The probability of a success is p = 0.55. The probability of a failure is q = .45. The number of trials is n = 20. The probability question can be stated mathematically as P(x = 15). If you define X as the number of losses, then a success is defined as a loss and a failure is defined as a win. A success does not necessarily represent a good outcome. It is simply the outcome that you are measuring. X still takes on the values of 0, 1, 2, 3, . . .,

PDF page 259: Read this as X is a random variable with a binomial distribution. The parameters are n and p: n = number of trials, p = probability of a success on each trial.