There are three characteristics of a binomial experiment.
- There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.
- The random variable, xx, number of successes, is discrete.
- There are only two possible outcomes, called "success" and "failure," for each trial. The letter p denotes the probability of a success on any one trial, and q denotes the probability of a failure on any one trial. p + q = 1.
- The n trials are independent and are repeated using identical conditions. Think of this as drawing WITH replacement. Because the n trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, p, of a success and probability, q, of a failure remain the same. For example, randomly guessing at a true-false statistics question has only two outcomes. If a success is guessing correctly, then a failure is guessing incorrectly. Suppose Joe always guesses correctly on any statistics true-false question with a probability p = 0.6. Then, q = 0.4. This means that for every true-false statistics question Joe answers, his probability of success (p = 0.6) and his probability of failure (q = 0.4) remain the same.
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.
The mean, \(\mu\), and variance, \(\sigma^2\), for the binomial probability distribution are \(\mu=n p\) and \(\sigma^2=n p q\). The standard deviation, \(\sigma\), is then \(\sigma=\sqrt{n p q}\).
Any experiment that has characteristics three and four and where \(n=1\) is called a Bernoulli Trial (named after Jacob Bernoulli who, in the late 1600s, studied them extensively). A binomial experiment takes place when the number of successes is counted in one or more Bernoulli Trials.
At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A "success" could be defined as an individual who withdrew. The random variable X = the number of students who withdraw from the randomly selected elementary physics class.
The state health board is concerned about the amount of fruit available in school lunches. Forty-eight percent of schools in the state offer fruit in their lunches every day. This implies that 52% do not. What would a "success" be in this case?
Suppose you play a game that you can only either win or lose. The probability that you win any game is \(55 \%\), and the probability that you lose is \(45 \%\). 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 \(0,1,2,3, \ldots, 20\). The probability of a success is \(p=0.55\). The probability of a failure is \(q=0.45\). The number of trials is \(n=20\). The probability question can be stated mathematically as \(P(x=15)\).
A trainer is teaching a rescued dolphin to catch live fish before returning it to the wild. The probability that the dolphin successfully catches a fish is 35%, and the probability that the dolphin does not successfully perform the trick is 65%. Out of 20 attempts, you want to find the probability that the dolphin succeeds 12 times. Find the P(X=12) using the binomial Pdf.
A coin has been altered to weight the outcome from 0.5 to 0.25 and flipped 5 times. Each flip is independent. What is the probability of getting more than 3 heads? Let X = the number of heads in 5 flips of the fair coin. X takes on the values 0, 1, 2, 3, 4, 5. Since the coin is altered to result in p = 0.25, q is 0.75. The number of trials is n = 5. State the probability question mathematically.
Solution
First develop fully the probability density function and graph the probability density function. With the fully developed probability density function we can simply read the solution to the question \(P(x>3)\) heads. \(P(x>3)=P(x=4)+P(x=5)=0.0146+0.0007=0.0153\). We have added the two individual probabilities because of the addition rule from Probability Topics.
Figure \(\PageIndex{1}\) also allows us to see the link between the probability density function and probability and area. We also see in Figure \(\PageIndex{1}\) the skew of the binomial distribution when p is not equal to 0.5 . In Figure 4.2 the distribution is skewed right as a result of \(\mu=n p=1.25\) because \(p=0.25\).
\[\begin{aligned}
P\left(x=x_0\right) & =\binom{n}{x} p^x(1-p)^{n-x} \\
= & \binom{5}{x_0} \cdot 25^{x_0} \cdot 75^{5-x_0} \\
& \text { etc. } \\
\mu= & \mathrm{np}=1.25
\end{aligned}\]
A fair, six-sided die is rolled ten times. Each roll is independent. You want to find the probability of rolling a one more than three times. State the probability question mathematically.
Approximately 70% of statistics students do their homework in time for it to be collected and graded. Each student does homework independently. In a statistics class of 50 students, what is the probability that at least 40 will do their homework on time? Students are selected randomly.
a. This is a binomial problem because there is only a success or a __________, there are a fixed number of trials, and the probability of a success is 0.70 for each trial.
b. If we are interested in the number of students who do their homework on time, then how do we define X?
c. What values does x take on?
d. What is a "failure," in words?
e. If p + q = 1, then what is q?
f. The words "at least" translate as what kind of inequality for the probability question P(x ____ 40).
- Answer
-
a. failure
b. \(X=\) the number of statistics students who do their homework on time
c. \(0,1,2, \ldots, 50\)
d. Failure is defined as a student who does not complete their homework on time.
The probability of a success is \(p=0.70\). The number of trials is \(n=50\).
e. \(q=0.30\)
f. greater than or equal to \((\geq)\)
The probability question is \(P(x \geq 40)\).
- Sixty-five percent of people pass the state driver’s exam on the first try. A group of 50 individuals who have taken the driver’s exam is randomly selected. Give two reasons why this is a binomial problem.