True or False? A binomial model cannot be used if the situation has more than two possible outcomes.

False. What matters is that each trial has two outcomes, not that the situation does. So, for example , a car can be many colours, but if the trial is is this car yellow? there are only two outcomes: yellow or not yellow. The other three conditions still have to hold.

A bag contains 6 caramels and 4 marshmallows, and a person eats 5 of the sweets. Someone wants to model the number of caramels eaten as a binomial distribution. Which binomial condition does this break, and why?

Independence , and with it the constant probability of success. Eating a caramel leaves fewer caramels in the bag, so the probability that the next sweet is a caramel depends on what has already been taken. Sampling without replacement from a small collection is the usual way this condition fails.

A LevelMathsEdexcelStatisticsFlashcardsStatistical DistributionsBinomial Distribution

Binomial Distribution (Edexcel A Level Maths: Statistics): Flashcards

Exam code: 9MA0

1/14

0Still learning

Know0

FrontThe Binomial Distribution
What four conditions must be satisfied for a situation to be modelled by a binomial distribution?
BackThe Binomial Distribution
There is a fixed number of trials, $n$
The trials are independent of each other
Each trial has exactly two outcomes, success or failure
The probability of success, $p$ , is constant
BackThe Binomial Distribution
Sign up to unlock flashcards
Join for free to unlock a full flashcard set, track what you know,
and turn revision into real progress.

Was this flashcard helpful?

Cards in this collection (14)

What four conditions must be satisfied for a situation to be modelled by a binomial distribution?
- There is a fixed number of trials, $n$
- The trials are independent of each other
- Each trial has exactly two outcomes, success or failure
- The probability of success, $p$ , is constant
What does the notation $X ~ B (20, 0.3)$ tell you about $X$ ?
$X ~ B (20, 0.3)$ tells you that $X$ counts the number of successes in 20 trials, where the probability of success in each trial is 0.3.
The first number in the bracket is always the number of trials, $n$ , and the second is the probability of success, $p$ .
For $X ~ B (n, p)$ , complete the mean and the variance:
$Mean =______, Variance =______$
The completed results are:
$Mean = n p, Variance = n p (1 - p)$
Take the square root of the variance to get the standard deviation.
Both are printed in the formula booklet.
True or False?
The vertical line graph of a binomial distribution is always symmetrical.
False.
It is symmetrical only when $p = 0.5$ .
When $p$ is close to 0 the graph has a tail to the right, and when $p$ is close to 1 it has a tail to the left. The closer $p$ is to 0.5, the more nearly symmetrical it becomes.
For $X ~ B (n, p)$ , let $Y$ be the number of failures in the same $n$ trials. How are $X$ and $Y$ related, and what distribution does $Y$ follow?
$X + Y = n$ , since every trial is either a success or a failure.
$Y$ is also binomial, with the two probabilities swapped:
$Y ~ B (n, 1 - p)$
What three things should you identify and state when setting up a binomial model?
- What a single trial is, for example checking one person's hair colour
- What counts as a success, for example having black hair
- The random variable, stated in full, for example let $X$ be the number of students in a class of 30 with black hair
Stating the variable in full is what turns the context into $X ~ B (30, p)$ .
True or False?
A binomial model cannot be used if the situation has more than two possible outcomes.
False.
What matters is that each trial has two outcomes, not that the situation does.
So, for example, a car can be many colours, but if the trial is is this car yellow? there are only two outcomes: yellow or not yellow. The other three conditions still have to hold.
A bag contains 6 caramels and 4 marshmallows, and a person eats 5 of the sweets. Someone wants to model the number of caramels eaten as a binomial distribution. Which binomial condition does this break, and why?
Independence, and with it the constant probability of success.
Eating a caramel leaves fewer caramels in the bag, so the probability that the next sweet is a caramel depends on what has already been taken.
Sampling without replacement from a small collection is the usual way this condition fails.
A random sample of 30 people is taken from a city in which 30% of the people have blue eyes. Strictly the trials are not independent, so why can a binomial model still be used?
Because the population is large.
Removing 30 people barely changes the proportion left, so each person can be treated as having an independent 30% chance of blue eyes, and $p$ is effectively constant.
A binomial model here rests on two assumptions: that the population is large, and that the sample is random.
For $X ~ B (n, p)$ , what is the formula for $P (X = x)$ , and why does it contain a binomial coefficient?
For $X ~ B (n, p)$ the formula is:
$P (X = x) = (\binom{n}{x}) p^{x} {(1 - p)}^{n - x}$
If there are $x$ successes then there are $n - x$ failures, which gives the two powers.
The coefficient $(\binom{n}{x})$ counts the number of different orders in which those $x$ successes can occur among the $n$ trials.
Your calculator has a binomial probability distribution function and a binomial cumulative distribution function. What does each one give you?
The probability distribution function gives $P (X = x)$ , the probability of exactly $x$ successes.
The cumulative distribution function gives $P (X \leq x)$ , the probability of $x$ successes or fewer.
Both need the same three inputs: the value of $x$ , the number of trials $n$ , and the probability of success $p$ .
For $X ~ B (n, p)$ , complete the identity used when your calculator only gives $P (X \leq x)$ :
$P (X \geq x) = 1 - P (X \leq______)$
The completed identity is:
$P (X \geq x) = 1 - P (X \leq x - 1)$
So, for example, $P (X \geq 10) = 1 - P (X \leq 9)$ .
The $- 1$ matters: subtracting $P (X \leq x)$ instead would wrongly remove $P (X = x)$ as well.
How do you find $P (4 \leq X \leq 9)$ for a binomial distribution using only $P (X \leq x)$ ?
Take everything up to 9 and remove everything below 4:
$P (4 \leq X \leq 9) = P (X \leq 9) - P (X \leq 3)$
In general $P (a \leq X \leq b) = P (X \leq b) - P (X \leq a - 1)$ .
Find the largest value you want to include and the largest value you want to exclude: those are the two numbers.
Why can $P (X < x)$ be rewritten as $P (X \leq x - 1)$ for a binomial distribution?
Because $X$ counts successes, so it takes only integer values, and the probability of any non-integer value is zero.
There is nothing between $x - 1$ and $x$ for the distribution to give probability to, so excluding $x$ is the same as stopping at $x - 1$ .
The same reasoning gives $P (X > x) = P (X \geq x + 1)$ .

Sign up to unlock flashcards

By signing up you agree to our Terms and Privacy Policy

Previous:Probability DistributionsNext:Normal Distribution