9.05 Binomial distributions
In a binomial experiment:
Are the trials independent or dependent?
How many possible outcomes are there for each trial?
State whether following scenarios describe a binomial random variable:
Rolling a six-sided die 20 times and counting the number of times the die lands on 1.
The sum of the outcomes of rolling a six-sided die 10 times.
Selecting 70 random people from the population and recording the number of females.
A particular coin is more likely to land on heads than tails. Tossing this coin 10 times and recording the number of times it lands on heads.
Drawing a marble with replacement from a bag containing purple, green and yellow marbles, and noting the number of purple marbles.
Drawing a marble with replacement from a bag containing purple, green and yellow marbles, and noting the number of red marbles.
Consider the binomial probability formula:P \left( x \right) = {}^{n}C_{x} p^{x} \left(1 - p\right)^{n - x}where P \left( x \right) is the probability of obtaining x successes in n independent trials.
What does p represent in the formula?
What does \left(1 - p\right) represent in the formula?
For a binomial probability distribution with n, trials and probability of success p, Ned performed the following calculation:\binom{n}{2} p^{2} \times 0.7^{4}
State the probability of success.
State the number of trials.
What probability is Ned trying to find with this calculation?
Consider the binomial power \left(p + q\right)^{4}.
Expand \left(p + q\right)^{4}.
We can interpret the terms in the expansion in terms of probabilities in a binomial distribution, with p the probability of success and q the probability of failure. What probability would the term \binom{4}{2} p^{2} q^{2} in the expansion represent?
Consider the three graphs below that show binomial distributions for n = 10 and p = 0.15, 0.5 and 0.85:
Which value of p corresponds to Graph A?
Which value of p corresponds to Graph B?
Which value of p corresponds to Graph C?
Consider a binomial distribution of a random variable X with p = 0.5 and n = 17.
Is P \left( \text{X = 5} \right) greater than, less than or equal to P \left( \text{X = 12} \right)?
Is the shape of the graph of this binomial distribution symmetric, positively skewed or negatively skewed?
Consider a binomial distribution of a random variable X with p = 0.13 and n = 11.
Is P \left( \text{X = 4} \right) greater than, less than or equal to P \left( \text{X = 7} \right)?
Is the shape of the graph of this binomial distribution symmetric, positively skewed or negatively skewed?
Consider a binomial distribution of a random variable X with p = 0.83 and n = 17.
Is P \left( \text{X = 2} \right) greater than, less than or equal to P \left( \text{X = 15} \right)?
Is the shape of the graph of this binomial distribution symmetric, positively skewed or negatively skewed?
For each of the followng table of values for the binomial distribution X:
Describe the shape of the graph as positively skewed, symmetric, or negatively skewed.
Calculate the value of p correct to two decimal places.
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| P\left(X=x\right) | 0.004 | 0.031 | 0.109 | 0.219 | 0.274 | 0.219 | 0.109 | 0.031 | 0.004 |
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| P\left(X=x\right) | 0.152 | 0.323 | 0.3 | 0.16 | 0.053 | 0.011 | 0.001 | 0 | 0 |
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| P\left(X=x\right) | 0 | 0 | 0.001 | 0.009 | 0.046 | 0.147 | 0.294 | 0.336 | 0.168 |
Find the value of the following:
A binomial random variable X has 13 independent trials and a probability of success of 0.4.
Using the definition for the binomial random variable X, write the formula for P \left( X = 2 \right).
Calculate P \left( X = 2 \right) to four decimal places.
For a binomial random variable X, P \left(X=3\right) = {}^{10}C_{3} \times 0.6^{3} \times 0.4^{7}.
Calculate P \left( X = 3 \right) to four decimal places.
State the number of trials in this experiment.
State the probability of success in this experiment.
X is a binomial variable with the probability mass function:P(X = k)={}^{4}C_{k} \times \left(0.4\right)^{k} \times \left(0.6\right)^{4 - k} \text{ for }k = 0, 1, 2, 3, 4
State the number of trials for this distribution.
State the probability of success.
Wrie an expression for P \left(X = 2 \right).
How many ways can we get 2 successes in the 4 trials?
Calculate the probability P \left(X = 2 \right).
X is a binomial variable with the probability mass function:P(X = k)={}^{3}C_{k} \times \left(0.3\right)^{k} \times \left(0.7\right)^{3 - k} \text{ for } k = 0, 1, 2, 3
Complete the table:
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P \left( X = x \right) | 0.343 | 0.441 |
Consider the random variable X, which has distribution X \sim B\left(4, 0.1 \right).
Write an expression for P \left(X = 1 \right).
Write an expression for P \left(X = 2 \right).
Complete the following table:
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P \left( X = x \right) | 0.6561 | 0.0001 |
A binomial random variable X is defined as:P \left(X=x\right) = {}^{20}C_{x} \times 0.75^{x} \times 0.25^{20 - x} \text{ for }x = 1, 2, \ldots, 20
State the number of trials in this experiment.
State the probability of success in this experiment.
Calculate P \left( X = 11 \right) to four decimal places.
X is a binomial variable with the probability mass function:P(X = k)={}^{6}C_{k} \times \left(0.8\right)^{k} \times \left(0.2\right)^{6 - k} \text{ for } k = 0, 1, 2, 3, 4, 5, 6
State the number of trials for this distribution.
What is the probability of failure?
Calculate the probability P \left(X = 3 \right).
Calculate the probability P \left(X \leq 3 \right).
X is a binomial variable with the probability mass function:P(X = k)={}^{5}C_{k} \times \left(0.4\right)^{k} \times \left(0.6\right)^{5 - k} \text{ for } k = 0, 1, 2, 3, 4, 5
Calculate P(X \geq 3).
Calculate P(X < 3).
X is a binomial variable with the probability mass function:P(X = k)={}^{5}C_{k} \times \left(0.2\right)^{k} \times \left(0.8\right)^{5 - k} \text{ for } k = 0, 1, 2, 3, 4, 5
State the number of trials for this distribution.
State the probability of success.
Calculate the probability P \left(X > 3 \right).
What is the most likely number of successes?
Calculate the mean of the distribution.
Calculate the variance of the distribution.
Consider the random variable X, which has distribution X \sim B\left(8, \dfrac{4}{5} \right).
State the number of trials for this distribution.
State the probability of success.
Calculate the probability P \left(X > 4 \right) to two decimal places.
What is the most likely number of successes?
Calculate the mean of the distribution.
Calculate the variance of the distribution.
X is a binomial variable with a probability distribution given in the table:
| x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| P ( X = x ) | 0.00243 | 0.02835 | 0.1323 | 0.3087 | 0.36015 | 0.16807 |
Find P(X \geq 4).
Find P(X < 3).
The following table represents a probability distribution of a binomial random variable:
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P \left( X = x \right) | 0.2401 | 0.4116 | 0.2646 | 0.0756 | 0.0081 |
State the number of trials for this distribution.
Calculate the probability P \left(X = 3 \right).
Calculate the probability P \left( X < 3 \right).
Using P \left(X = 4 \right), calculate the probability of success p.
Calculate E \left( X \right).
Calculate the standard deviation of the distribution.
The table represents a cumulative probability distribution of a binomial random variable:
| x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| P \left( X \leq x \right) | 0.01024 | 0.08704 | 0.31744 | 0.66304 | 0.92224 | 1 |
State the number of trials for this distribution.
Calculate P \left(X \leq 2 \right).
Calculate P \left( X \geq 3 \right).
Calculate P \left( X = 5 \right).
Using P \left(X = 5 \right), calculate the probability of success p.
Calculate the standard deviation of the distribution.
The following graph represents a probability distribution of a binomial random variable:
State the number of trials for this distribution.
Calculate P \left(X = 3 \right).
Calculate P \left( X < 3 \right).
Using P \left(X = 4 \right), calculate the probability of success p.
Calculate E \left( X \right).
Calculate the standard deviation of the distribution.
The following graph represents a cumulative probability distribution of a binomial random variable:
State the number of trials for this distribution.
Calculate P \left(X \leq 2 \right).
Calculate P \left( X \geq 3 \right).
Calculate P \left( X = 5 \right).
Using P \left(X = 5 \right), calculate the probability of success p.
Calculate the standard deviation of the distribution.
Calculate the probability of the following, rounding your answers to two decimal places:
Getting exactly 18 successes in 30 independent trials if the probability of success in each trial is 0.7.
Getting at least 10 successes in 80 independent trials if the probability of success in each trial is p = 0.1.
Getting at most 22 successes in 25 independent trials if the probability of success in each trial is 0.85.
Getting more than 15 successes in 33 independent trials if the probability of success in each trial is 0.45.
Getting less than 12 successes in 50 independent trials if the probability of success in each trial is 0.28.
P \left(X = 2 \right) = 0.3456 for a binomial random variable X with 4 trials.
If success is more likely, find the probability of success p.
Calculate E \left(X\right).
Calculate \text{Var} \left(X\right).
E \left( X \right) = 6 for a binomial random variable X with 10 trials.
Find the probability of success p.
Calculate \text{Var} \left(X\right).
The expected value of a binomial distribution is 8 and the variance is 6.4.
Find the number of trials, n.
Find the probability of success, p.
E \left( X \right) = \dfrac{15}{2} for a binomial random variable X with probability of success \dfrac{3}{10}.
Find the number of trials n.
Calculate the standard deviation of X.
E \left(X\right) = 2 and \text{Var} \left(X\right) = 1.6 for a binomial random variable X.
Find the probability of success p.
Find the number of trials n.
E \left(X\right) = 4.8 and \sigma \left(X\right) = \sqrt{2.88} for a binomial random variable X.
Find the probability of success p.
Find the number of trials n.
A certain disease has a survival rate of 64\%. Of the next 110 people who contract the disease, how many would you expect to survive? Round your answer to the nearest whole number.
The probability of a particular tennis player getting their first serve in is 0.72. If she serves 90 times in a match, how many times would she expect to get her first serve in? Round your answer rounded to the nearest whole number.
A fair die is rolled 5 times and the number of fours noted.
On each trial, what is the probability of success?
Calculate the probability of rolling exactly 2 fours, correct to four decimal places.
Calculate the probability of rolling exactly 3 fours, correct to four decimal places.
David owns 3 yellow mugs and 8 blue mugs. Each morning, he chooses a mug at random for his cup of tea before work, 5 days a week. The mug is washed and returned before the next day.
Each day, what is the probability of choosing a yellow mug?
What is the probability that he used a yellow mug exactly twice this past work week? Round your answer to four decimal places.
What is the probability that he never used a yellow mug this past work week? Round your answer to four decimal places.
At a local Italian restaurant, 16 people are in line to order. The probability that any one person will choose pizza is 23\%.
What is the probability that exactly 4 people choose pizza? Round your answer to four decimal places.
What is the probability that exactly half of the people choose pizza? Round your answer to four decimal places.
Consider the binomial \left(\dfrac{1}{5} + \dfrac{4}{5}\right)^{5}.
Write the formula for the term of the expansion that contains \left(\dfrac{4}{5}\right)^{3}.
A box of chocolates is \dfrac{1}{5} dark chocolate and \dfrac{4}{5} milk chocolate. If 5 chocolates are selected at random with replacement, describe what the term from part (a) would tell us.
Calculate the probability of selecting exactly 1 dark chocolate.
Calculate the probability of selecting at most 1 dark chocolate.