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Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
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📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
A random variable is a variable that can take different values, each associated with some probability, arising from a random process or phenomenon. We usually denote random variables with a capital letter, such as X.
It can be discrete (taken from a finite set of values) or continuous (taking any value within an interval).
The probability distribution of a random variable tells us how likely each outcome is.
Examples
Human height
The sum of a two dice roll
The number of goals scored in a game
A discrete random variable takes from a finite set of values:
where each possible value has an associated probability.
The sum of the probabilities for all possible values {x1,x2,…xn} of a discrete random variable X equals 1. In symbols,
where U denotes the sample space.
Probability distributions of discrete random variables can be given in a table or as an expression. As a table, distributions have the form
where the values in the row P(X=x) sum to 1.
Probability distributions can be given in a table or as an expression. As an expression, distributions have the form
for any discrete random variable X.
The expected value of a discrete random variable X is the average value you would get if you carried out infinitely many repetitions. It is a weighted sum of all the possible values:
The expected value is often denoted μ.
In probability, a game is a scenario where a player has a chance to win rewards based on the outcome of a probabilistic event. The rewards that a player can earn follow a probability distribution, for example X, that governs the likelihood of winning each reward. The expected return is the reward that a player can expect to earn, on average. It is given by E(X), where X is the probability distribution of the rewards.
Games can also have a cost, which is the price a player must pay each time before playing the game. If the cost is equal to the expected return, the game is said to be fair.
The binomial distribution models situations where the same action is repeated multiple times, each with the same chance of success. It has two key numbers: the number of attempts (n) and the probability of success in each attempt (p).
If a random variable X follows a binomial distribution, we write X∼B(n,p).
The binomial probability density function (aka pdf) is a function that models the likelihood of obtaining k successes from n trials where the likelihood of success of each trial is p. We calculate the probability of exactly k successes in n trials, P(X=k), using the calculator's binompdf function.
Press 2nd → distr → binompdf(. Once in the binompdf function, write your n value after "trials," your p value after "p," and your k value after "x value." Then hit enter twice and the calculator will return the probability you are interested in.
The distr button is located above vars . Once in the distr menu, you can also click alpha → A to navigate to the binompdf function.
The binomial cumulative density function tells us the probability of obtaining k or fewer successes in n trials, each with a likelihood of success of p. We calculate the probability of less than or equal to k successes in n trials, P(X≤k), using the calculator's binomcdf function.
Press 2nd → distr → binomcdf(. Once in the binomcdf function, write your n value after "trials," your p value after "p," and your k value after "x value." Then hit enter twice and the calculator will return the probability you are interested in.
The distr button is located above vars . Once in the distr menu, you can also click alpha → B to navigate to the binomcdf function.
A student is taking a 20 question multiple choice exam where each question is worth 1 point. The student needs to score 11 points for a 5, and 15 points for a 6.
Given that the probability the student answers each question correctly is 0.6, find the probability that he scored a 5.
Let X∼B(20,0.6) be the student's score. The student scores a 5 if 11≤X<15 (i.e. 11≤X≤14). We can express this probability as the difference of two probabilities:
Using a calculator, we find
P(X≤14)=binomcdf(20, 0.6, 14)=0.874401
P(X≤10)=binomcdf(20, 0.6, 10)=0.244663
Subtracting we find P(11≤X≤14)=0.630.
Note that you get the same result from doing binompdf(20, 0.6, 11) + binompdf(20, 0.6,12) + binompdf(20, 0.6, 13) + binompdf(20, 0.6, 14)
We could visualize this on a graph as
If X∼B(n,p), then
and
The normal distribution, often called the bell curve, is a symmetric, bell-shaped probability distribution widely used to model natural variability and measurement errors. It appears frequently in natural settings because averaging many small, independent effects tends to produce results that cluster around a central value, naturally forming a bell-shaped distribution.
The normal distribution is characterized by its mean, μ, and standard deviation, σ, which completely and uniquely describe both the central value and how "spread out" the curve is. By convention, we describing the normal distribution by writing X∼N(μ,σ2). Notice that σ2 is the variance, not the standard deviation.
The probability that X is less than a given value a, written P(X<a), is equal to the area under the curve to the left of x=a:
It follows that the total area under the curve is 1, which is required as the probabilities must sum to 1.
Because of the symmetry of the normal distribution, we know that
Further, for any real number a,
It is also useful (but not often required) to know the empirical rule:
To calculate P(a<X<b) for X∼N(μ,σ2) on your GDC, press 2nd →distr (on top of vars) to open the probability distribution menu. Select normalcdf( with your cursor. Type the value of a after "lower," the value of b after "upper," the value of μ after "μ," and the value of √σ2=σ after σ (since the calculator asks for standard deviation, not variance). Then click enter twice and the calculator will return the value of P(a<X<b).
If you want to find a one-sided probability like P(a<X), enter the value ±1×1099 as the upper or lower bound.
Under the hood, the calculator is finding the area under the normal curve between x=a and x=b:
The calculator can also perform inverse normal calculations. That is, given the mean μ, the standard deviation σ, and the probability P(X<a)=k, the calculator can find the value a.
On your GDC, press 2nd →distr (on top of vars) to open the probability distribution menu. Select invNorm( with your cursor. Type the value of k after "area," the value of μ after "μ," and the value of √σ2=σ after σ (since the calculator asks for standard deviation, not variance). Then click enter twice and the calculator will return the value of a.
Note the calculator specifically returns the value of the "left end" of the tail. To find the value of some b when given P(b<X)=k, enter the value of 1−k (the complement) as the area.
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
A random variable is a variable that can take different values, each associated with some probability, arising from a random process or phenomenon. We usually denote random variables with a capital letter, such as X.
It can be discrete (taken from a finite set of values) or continuous (taking any value within an interval).
The probability distribution of a random variable tells us how likely each outcome is.
Examples
Human height
The sum of a two dice roll
The number of goals scored in a game
A discrete random variable takes from a finite set of values:
where each possible value has an associated probability.
The sum of the probabilities for all possible values {x1,x2,…xn} of a discrete random variable X equals 1. In symbols,
where U denotes the sample space.
Probability distributions of discrete random variables can be given in a table or as an expression. As a table, distributions have the form
where the values in the row P(X=x) sum to 1.
Probability distributions can be given in a table or as an expression. As an expression, distributions have the form
for any discrete random variable X.
The expected value of a discrete random variable X is the average value you would get if you carried out infinitely many repetitions. It is a weighted sum of all the possible values:
The expected value is often denoted μ.
In probability, a game is a scenario where a player has a chance to win rewards based on the outcome of a probabilistic event. The rewards that a player can earn follow a probability distribution, for example X, that governs the likelihood of winning each reward. The expected return is the reward that a player can expect to earn, on average. It is given by E(X), where X is the probability distribution of the rewards.
Games can also have a cost, which is the price a player must pay each time before playing the game. If the cost is equal to the expected return, the game is said to be fair.
The binomial distribution models situations where the same action is repeated multiple times, each with the same chance of success. It has two key numbers: the number of attempts (n) and the probability of success in each attempt (p).
If a random variable X follows a binomial distribution, we write X∼B(n,p).
The binomial probability density function (aka pdf) is a function that models the likelihood of obtaining k successes from n trials where the likelihood of success of each trial is p. We calculate the probability of exactly k successes in n trials, P(X=k), using the calculator's binompdf function.
Press 2nd → distr → binompdf(. Once in the binompdf function, write your n value after "trials," your p value after "p," and your k value after "x value." Then hit enter twice and the calculator will return the probability you are interested in.
The distr button is located above vars . Once in the distr menu, you can also click alpha → A to navigate to the binompdf function.
The binomial cumulative density function tells us the probability of obtaining k or fewer successes in n trials, each with a likelihood of success of p. We calculate the probability of less than or equal to k successes in n trials, P(X≤k), using the calculator's binomcdf function.
Press 2nd → distr → binomcdf(. Once in the binomcdf function, write your n value after "trials," your p value after "p," and your k value after "x value." Then hit enter twice and the calculator will return the probability you are interested in.
The distr button is located above vars . Once in the distr menu, you can also click alpha → B to navigate to the binomcdf function.
A student is taking a 20 question multiple choice exam where each question is worth 1 point. The student needs to score 11 points for a 5, and 15 points for a 6.
Given that the probability the student answers each question correctly is 0.6, find the probability that he scored a 5.
Let X∼B(20,0.6) be the student's score. The student scores a 5 if 11≤X<15 (i.e. 11≤X≤14). We can express this probability as the difference of two probabilities:
Using a calculator, we find
P(X≤14)=binomcdf(20, 0.6, 14)=0.874401
P(X≤10)=binomcdf(20, 0.6, 10)=0.244663
Subtracting we find P(11≤X≤14)=0.630.
Note that you get the same result from doing binompdf(20, 0.6, 11) + binompdf(20, 0.6,12) + binompdf(20, 0.6, 13) + binompdf(20, 0.6, 14)
We could visualize this on a graph as
If X∼B(n,p), then
and
The normal distribution, often called the bell curve, is a symmetric, bell-shaped probability distribution widely used to model natural variability and measurement errors. It appears frequently in natural settings because averaging many small, independent effects tends to produce results that cluster around a central value, naturally forming a bell-shaped distribution.
The normal distribution is characterized by its mean, μ, and standard deviation, σ, which completely and uniquely describe both the central value and how "spread out" the curve is. By convention, we describing the normal distribution by writing X∼N(μ,σ2). Notice that σ2 is the variance, not the standard deviation.
The probability that X is less than a given value a, written P(X<a), is equal to the area under the curve to the left of x=a:
It follows that the total area under the curve is 1, which is required as the probabilities must sum to 1.
Because of the symmetry of the normal distribution, we know that
Further, for any real number a,
It is also useful (but not often required) to know the empirical rule:
To calculate P(a<X<b) for X∼N(μ,σ2) on your GDC, press 2nd →distr (on top of vars) to open the probability distribution menu. Select normalcdf( with your cursor. Type the value of a after "lower," the value of b after "upper," the value of μ after "μ," and the value of √σ2=σ after σ (since the calculator asks for standard deviation, not variance). Then click enter twice and the calculator will return the value of P(a<X<b).
If you want to find a one-sided probability like P(a<X), enter the value ±1×1099 as the upper or lower bound.
Under the hood, the calculator is finding the area under the normal curve between x=a and x=b:
The calculator can also perform inverse normal calculations. That is, given the mean μ, the standard deviation σ, and the probability P(X<a)=k, the calculator can find the value a.
On your GDC, press 2nd →distr (on top of vars) to open the probability distribution menu. Select invNorm( with your cursor. Type the value of k after "area," the value of μ after "μ," and the value of √σ2=σ after σ (since the calculator asks for standard deviation, not variance). Then click enter twice and the calculator will return the value of a.
Note the calculator specifically returns the value of the "left end" of the tail. To find the value of some b when given P(b<X)=k, enter the value of 1−k (the complement) as the area.