# Binomial Probabilities Examples and Questions

In a binomial experiment, you have a number \( n \) of independent trials and each trial has two possible outcomes or several outcomes that may be reduced to two outcomes.

The properties of a binomial experiment are:

1) The number of trials \( n \) is constant.

2) Each trial has 2 outcomes (or that can be reduced to 2 outcomes) only: "success" or "failure" , "true" or "false", "head" or "tail", ...

3) The probability \( p \) of a success in each trial must be constant.

4) The outcomes of the trials must be independent of each other.

Examples of binomial experiments

1) Toss a coin \( n = 10 \) times and get \( k = 6 \) heads (success) and \( n - k \) tails (failure).

2) Roll a die \( n = 5\) times and get \( 3 \) "6" (success) and \( n - k \) "no 6" (failure).

3) Out of \( n = 10 \) tools, where each tool has a probability \( p \) of being "in good working order" (success), select 6 at random and get 4 "in good working order" and 2 "not in working order" (failure).

4) A newly developed drug has probability \( p \) of being effective.

Select \( n \) people who took the drug and get \( k \) "successful treatment" (success) and \( n - k \) "not successful treatment" (failure).

## Binomial Formula ExplanationsThe best way to explain the formula for the binomial distribution is to solve the following example.
Example 1
## Mean and Standard Deviation of a Binomial DistributionMean: \( \mu = n \cdot p \) , Standard Deviation: \( \sigma = \sqrt{ n \cdot p \cdot (1-p)} \)## Examples on the Use of the Binomial FormulaMore examples and questions on how the binomial formula is used to solve probability questions and solve problems.
Example 2
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Example 7
Example 8
## Questions and their Solutions
## Question 1A die is rolled 5 times.a) Find the probability 3 even numbers are obtained. b) Find the probability that at least 3 even numbers are obtained. c) Find the probability that at most 3 even numbers are obtained.
## Question 2A card is drawn from a deck of 52 cards at random, its color noted and then replaced back into the deck, 10 times.a) Find the probability of getting at least 3 red cards.
## Question 3A multiple choice test has 20 questions. Each question has five possible answers with one correct answer per question. What is the probability that a student will answer 15 or more questions correct (to pass) by guessing randomly?. Conclusion.
## Question 4According to an OCDE report (https://data.oecd.org/eduatt/population-with-tertiary-education.htm); for the age group between 25 and 34 years, 61.8% in Canada and 50.8% in the United Kingdom have a tertiary education.If 200,000 people, in the age between 25 and 34 years, are selected at random in Canada and 200,000 in the same age group are selected at random in the United kingdom, how many are expected to have tertiary education in each of these two countries? ## Solutions to the Above Questions## Solution to Question 1
a)
## Solution to Question 2
Because the card is replaced back, it is a binomial experiment with the number of trials \( n = 10 \)
## Solution to Question 3
Each question has 5 possible answers with one correct. Therefore the probability of getting a correct answer in one trial is \( p = 1/5 = 0.2 \)
## Solution to Question 4
In both cases, it is a binomial experiment with
## More References and linksBinomial Probability Distribution Calculatoraddition rule of probabilities multiplication rule of probabilities probability questions classical formula for probability mutually exclusive events Introduction to Probabilities sample space event elementary statistics and probabilities. Home Page |