The Binomial Experiment and the Binomial Formula (6.5)

Summary

This video script offers an insightful exploration of the binomial setting and formula within the context of probability distribution. It explains the concept of binomial experiments, which involve fixed trials with two outcomes, constant success probability, and independent trials. Through examples like coin flipping and marble drawing, the script demonstrates how to calculate probabilities of specific outcomes and verifies binomial conditions. It also introduces the binomial formula as a shortcut for these calculations, emphasizing its applicability to binomial experiments only.

Takeaways

• 📚 The binomial setting and formula are discussed, focusing on the probability of success or failure in repeated experiments.
• 🔢 The prefix 'bi' signifies two outcomes, such as success or failure, in binomial probabilities.
• 🧩 Four conditions must be met for a binomial setting: fixed number of trials, two outcomes per trial, constant probability of success, and independence of trials.
• 🪙 An example of a binomial experiment is flipping a coin multiple times, calculating the probability of getting a certain number of heads.
• 🎲 The probability of getting exactly one head in three coin flips is calculated by considering all possible outcomes and their probabilities.
• 📉 Each outcome's probability is calculated using the product of the probabilities of individual flips, which in this case is 0.125.
• 📈 The total probability of getting exactly one head is the sum of the probabilities of all individual outcomes, resulting in 0.375.
• 🟢 The script checks if the coin flip experiment is binomial by confirming it satisfies all four binomial conditions.
• 🔄 Another example involves drawing marbles with replacement from a box, illustrating the binomial setting with a different context.
• 🎯 The probability of drawing exactly two green marbles from a box of ten is calculated using both direct enumeration and the binomial formula.
• 📝 The binomial formula provides a shortcut for calculating probabilities in binomial experiments, represented as 'n choose k' times the success probability raised to the power of successes, times the failure probability raised to the power of failures.

Q & A

What is the binomial probability distribution?

The binomial probability distribution refers to the probability of a success or failure in an experiment that is repeated multiple times, resulting in two possible outcomes for each trial.

What does the prefix 'bi' signify in the term 'binomial'?

The prefix 'bi' signifies two, as seen in words like 'bicycle' and 'binoculars', and in the context of binomial probabilities, it refers to the two outcomes: a success or a failure.

What are the four conditions that must be satisfied for a setting to be considered binomial?

The four conditions are: 1) a fixed number of trials, 2) only two possible outcomes for each trial, 3) the probability of success must be constant for every trial, and 4) each trial must be independent of the others.

Can you provide an example of a binomial experiment?

An example of a binomial experiment is flipping a regular coin three times, where the probability of getting heads (success) or tails (failure) remains constant and each flip is independent.

How many possible ways are there to get exactly one head when flipping a coin three times?

There are three possible ways to get exactly one head when flipping a coin three times: H-T-T, T-H-T, and T-T-H.

What is the probability of getting exactly one head when flipping a coin three times?

The probability of getting exactly one head when flipping a coin three times is 0.375, which is calculated by adding the probabilities of the three different outcomes (0.125 each).

What does it mean for the trials in a binomial experiment to be independent?

For trials to be independent in a binomial experiment means that the outcome of one trial does not influence the outcome of another trial.

How does the binomial formula help in calculating probabilities in a binomial experiment?

The binomial formula provides a shortcut for calculating probabilities by using the number of trials, the number of successes, and the probability of success, without having to list all possible outcomes.

What is the probability of drawing exactly two green marbles from a box of 10 marbles with 3 pink, 2 green, and 5 blue marbles when drawing 5 marbles with replacement?

The probability is 0.2048, calculated using the binomial formula or by adding the probabilities of all possible outcomes where exactly two green marbles are drawn.

What is the binomial formula and how is it structured?

The binomial formula is structured as P(k) = n choose k * (p^k) * ((1-p)^(n-k)), where k is the number of successes, n is the number of trials, p is the probability of success, and 'n choose k' is the combination formula representing the number of ways to choose k successes from n trials.

Why is it important to check if all four conditions are satisfied before applying the binomial formula?

It is important to ensure that all four conditions are satisfied to confirm that the experiment is a binomial experiment, as the binomial formula can only be applied to binomial settings.

Outlines

00:00

📚 Introduction to Binomial Setting and Formula

This paragraph introduces the concept of the binomial setting and formula in the context of probability. It explains that the binomial probability distribution is about the likelihood of success or failure in repeated experiments. The prefix 'bi' is highlighted to emphasize the two possible outcomes. The binomial setting is defined by four conditions: a fixed number of trials, two possible outcomes per trial, constant probability of success, and independence of trials. An example of flipping a coin three times is used to illustrate these concepts, with a step-by-step explanation of how to calculate the probability of getting exactly one head. The paragraph concludes by confirming that the coin flip scenario is a binomial experiment due to the satisfaction of all four conditions.

05:02

🎲 Applying the Binomial Setting to a Marble Drawing Experiment

The second paragraph delves into a more complex example involving drawing marbles from a box with replacement. It first establishes whether the scenario meets the binomial setting criteria, confirming that it does due to a fixed number of trials, two outcomes (success or failure), constant probability of success, and independent trials. The probability of drawing a green marble is calculated as 0.2, and the probability of not drawing a green marble as 0.8. The paragraph then explores the different ways of drawing exactly two green marbles out of five trials, emphasizing the uniform probability of 0.2048 for each of the ten possible outcomes. It concludes by demonstrating the use of the binomial formula as a shortcut for calculating the probability of exactly two successes in a binomial experiment, yielding the same result as the manual calculation.

Keywords

💡Binomial Setting

The 'Binomial Setting' refers to the conditions required for an experiment to be classified as binomial. It is integral to the video's theme as it lays the foundation for understanding binomial probability distributions. In the script, the binomial setting is explained through four conditions: a fixed number of trials, two possible outcomes (success or failure), a constant probability of success for each trial, and the independence of trials. An example of flipping a coin three times is used to illustrate this concept.

💡Binomial Probability Distribution

This term describes the probability of obtaining a certain number of successes in a fixed number of independent trials, each with two possible outcomes. The video's main theme revolves around this concept, explaining it through the binomial setting and formula. The script provides a practical example of flipping a coin to demonstrate how binomial probability distribution is calculated.

💡Success and Failure

In the context of the video, 'success' and 'failure' are the two possible outcomes in a binomial experiment. These terms are fundamental to the video's message as they define the nature of the outcomes in a binomial setting. For instance, in the coin flipping example, 'success' is defined as getting heads, while 'failure' is getting tails.

💡Binomial Experiment

A 'Binomial Experiment' is an experiment that meets the criteria of the binomial setting. The video uses this term to describe situations where the binomial probability distribution is applicable. The script illustrates this with examples such as flipping a coin and drawing marbles from a box, both of which satisfy the four conditions of a binomial setting.

💡Probability of Success

The 'Probability of Success' is the likelihood of achieving a 'success' outcome in a single trial of a binomial experiment. It is a key concept in the video, as it is a constant value that influences the binomial probability calculations. The script mentions that this probability must remain constant for every trial, as seen in the coin flipping and marble drawing examples.

💡Independence

The term 'Independence' in the video refers to the condition that the outcome of one trial in a binomial experiment does not affect the outcomes of other trials. This concept is crucial to the video's theme, as it ensures that each trial's result is not influenced by previous ones. The script explains this with the marble drawing example, emphasizing that replacing the marble after each draw maintains trial independence.

💡Combination Formula

The 'Combination Formula', often represented as 'n choose k', is used in the binomial formula to calculate the number of ways a certain number of successes can occur in a given number of trials. The video introduces this formula as part of the binomial formula explanation. It is used in the script to calculate the number of ways to draw exactly two green marbles from a box of ten marbles.

💡Binomial Formula

The 'Binomial Formula' is a mathematical formula used to calculate the probability of obtaining exactly k successes in n independent trials, each with the same probability of success. This formula is central to the video's content, as it provides a method to compute binomial probabilities efficiently. The script demonstrates its application in the marble drawing example.

💡Trials

'Trials' in the video refers to the individual attempts or instances within a binomial experiment. The concept is essential to the video's theme, as the number of trials determines the scope of the experiment. The script uses the term in the context of flipping a coin three times and drawing five marbles, where each action represents a single trial.

💡Probability Calculation

The term 'Probability Calculation' encompasses the process of determining the likelihood of an event occurring, which is a key focus of the video. The script demonstrates this through examples, showing how to calculate the probability of getting exactly one head in a coin flip and exactly two green marbles in a series of draws.

💡Supporting the Video

While not a technical term, 'Supporting the Video' is a call to action mentioned in the script, encouraging viewers to support the creators on Patreon and visit their website for more learning resources. This phrase is part of the video's conclusion, serving as a way to engage the audience beyond the educational content.

Highlights

Introduction to the binomial setting and formula in probability distribution.

Explanation of the binomial probability distribution involving success or failure in repeated experiments.

The prefix 'bi' signifies two outcomes: success or failure in binomial probabilities.

Four conditions required for the binomial setting: fixed number of trials, two outcomes, constant probability of success, and independence of trials.

Example of a binomial experiment: flipping a coin three times to find the probability of getting exactly one head.

Calculating probabilities by considering different outcomes and their respective probabilities.

Verification of a binomial experiment by checking if it satisfies the four binomial conditions.

Demonstration of calculating the probability of drawing exactly two green marbles from a box of 10 marbles with replacement.

Use of the combination formula in scientific calculators to simplify binomial probability calculations.

Explanation of how to determine the probability of a success and a failure in a binomial setting.

Calculation of the probability of drawing exactly two green marbles using both direct calculation and the binomial formula.

The binomial formula provides a shortcut for calculating probabilities in binomial experiments.

Emphasizing the importance of ensuring a binomial setting before applying the binomial formula.

Practical application of the binomial formula to solve the marble drawing problem.

Final calculation and verification of the probability using the binomial formula.

Encouragement to support the creators for more educational content on Patreon and the website.

Transcripts

in this video we'll be learning about

the binomial setting and the binomial

formula when we talk about the binomial

probability distribution we are

referring to the probability of a

success or a failure in an experiment

that is repeated multiple times you can

easily remember this by paying attention

to the prefix bi which literally means

two for example in bicycles there are

two wheels and in binoculars there are

two lenses

however in binomial probabilities there

are two outcomes a success or a failure

before we do some practice problems we

need to talk about the binomial setting

the binomial setting must satisfy four

conditions the first condition is that

the number of trials n must be fixed the

second condition is that there are only

two possible outcomes for each trial you

can either have a success or a failure

the third condition is that the

probability of success must be constant

for every trial and finally the fourth

condition is that each trial must be

independent this means that the outcome

of one trial does not influence the

outcome of another trial an experiment

that satisfies these four conditions is

called a binomial experiment the

binomial setting will make more sense

after we do an example so let's jump

into it if you flip a regular coin three

times what is the probability of getting

exactly one head and is this a binomial

experiment feel free to pause the video

at this point so you can try this

question for yourself

since the question says that we are

flipping the coin three times I will

have three blank spaces one for each

flip the first blank space is for the

first flip the second is for the second

flip and the third is for the third flip

in order to solve this problem we have

to realize that there are only three

possible ways for us to get exactly one

head the first way is to get heads on

the first flip and then tails on the

second and third flip the second way is

to get tails on the first flip heads on

the second flip and then tails on the

third flip the third and final way is to

get tails on the first and second flip

and then getting heads on the third flip

so overall we see that there are three

different outcomes where we get exactly

one head and we see that they vary based

on the order now all we have to do is

calculate the probabilities of each

outcome and add them together to get the

answer we know that the probability for

getting heads is 0.5 and the probability

of getting tails is also 0.5 to

calculate the probability of the first

outcome we will multiply the probability

of heads times the probability of tails

times the probability of tails again

this is equal to 0.5 times 0.5 times 0.5

and this is equal to 0.125 when we

calculate the probabilities for the

second and third outcome you'll find

that they will also be equal to 0.125

now all we have to do is add these

probabilities together to get the answer

and when we do we get the probability of

getting exactly one head which is equal

to 0.375

to check if this is a binomial

experiment we have to see if it

satisfies the for binomial conditions

the first condition is satisfied because

we have a fixed number of trials n is

equal to 3 because the experiment has to

be repeated 3 times in other words the

coin has to be flipped three times the

second condition is also satisfied

because we can define a success as

getting heads and we can define a

failure as not getting heads this is the

same as saying getting tails is a

failure

the third condition is also satisfied

because the probability of success

remained constant for every trial in

other words the probability of getting

heads is 0.5 and it stayed that way

every time the coin was flipped and

finally the fourth condition is also

satisfied because each trial is

independent gaining heads or tails for

one trial doesn't change the probability

of getting heads or tails for the other

trials since all four conditions are

satisfied we know that this is a

binomial experiment now let's do a

harder problem suppose we have 10

marbles in a box we have three pink

marbles two green marbles and five blue

marbles if we pick all five marbles with

replacement what is the probability of

drawing exactly two green marbles and is

this a binomial experiment feel free to

pause the video at this point so you can

try this question for yourself

before we calculate any probabilities

let's see if we have a binomial setting

is there a fixed number of trials yes n

is equal to 5 because we are picking out

5 marbles are the two possible outcomes

a success and a failure yes a success is

getting a green marble and a failure is

not getting a green marble

is the probability of success constant

for each trial yes because we are doing

the experiment with replacement every

time we conduct a trial or randomly pick

out one marble we put it back into the

box before drawing another marble if we

didn't do this the trials would become

dependent on one another instead of

being independent in this case the

probability of success is equal to the

probability of getting one green marble

which is equal to two over ten or 0.2

our trials independent of each other for

the reasons mentioned previously the

answer is yes since all four conditions

are satisfied we know that this is the

binomial experiment to solve this

problem we need to write out all the

possible ways of drawing exactly two

green marbles for example one way is

drawing a green marble on the first and

second try and then not drawing a green

marble on the third fourth and fifth try

another way could be only drawing a

green marble on the second and fifth try

notice that I decided to write a dashed

line for not getting a green marble this

is because I don't care if the marble

was blue or pink if I don't get a green

marble it's a failure in a binomial

setting we only care if we got a failure

or a success overall we see that there

are ten different ways of drawing two

green marbles we can properly reframe

this by saying that there are ten

different ways of getting two successes

and three failures now let's calculate

some probabilities the probability of a

success is equal to the probability of

drawing a green marble since there are

ten marbles in the box and only two of

them are green the probability of

drawing a green marble is equal to two

over ten or 0.2 therefore the

probability of a success is equal to 0.2

the probability of a failure is equal to

their probability of not drawing a green

marble there are 10 marbles in the box

and eight of them aren't green so the

probability of not drawing a green

marble is just 8 over 10 or 0.8

therefore the probability of a failure

is equal to 0.8 let's calculate the

probability of the first outcome we have

two successes followed by three failures

so we will have 0.2 times 0.2 times 0.8

times 0.8 times 0.8 which gives us an

answer of zero point zero two zero four

eight if we do a similar calculation for

the rest of the outcomes you should also

get an answer of zero point zero two

zero four eight notice how each of these

ten outcomes has the same probability

the reason for this is because each

outcome has two successes and three

failures so it makes sense that they all

have the same probabilities now to

calculate the probability of drawing

exactly two green marbles all we have to

do is add up all these probabilities

together and when we do we get an answer

of zero point two zero four eight

however there is another way we can

calculate this answer and that is by

using the binomial formula the binomial

formula looks like this it looks a

little scary but let's break it down k

is the number of successes n is the

number of trials and little P is the

probability of success what we have here

in brackets represents the combination

formula it is often seen in scientific

calculators as NCR which is the same

thing as saying n choose R so for the

binomial formula we say that the

probability of k is equal to n choose k

times the probability of success little

P raised to the number of successes K

times the probability of failure which

is 1 minus P raised to the number of

failures which is equal to n minus K

by using this formula we are essentially

giving ourselves a nice shortcut for

calculating things however it's

important to remember that we can only

use this formula if we have a binomial

experiment let's go back to the problem

so I can show you how to use this

formula if we pickle five marbles with

replacement what is the probability of

drawing exactly two green marbles in

this problem and is equal to five since

there are five trials in other words we

are randomly picking out five marbles K

is equal to two since we are only

concerned about getting exactly two

successes this is the same as getting

exactly two green marbles little P is

equal to zero point two as this is the

probability of success that we

calculated earlier now all we have to do

is plug these values into the formula

and we'll get an answer so the

probability of drawing exactly two green

marbles in other words the probability

of getting two successes is equal to 5

choose 2 times 0.2 squared times 1 minus

0.2 raised to the power of 5 minus 2

after this calculation we get an answer

of zero point two zero four eight which

is the same answer that we calculated

from before

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