Tutorial: Bernoulli trials and binomial random variables

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(This topic is also in Section 9.2 in Finite Mathematics and Applied Calculus)

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Resources

Binomial distribution utility

Histogram utility

The binomial random variable

%%Note To follow this tutorial, you need to know what a random variable is. Go to the %%prevsectut to review that if necessary.

In many experiments there are only two outcomes whose probabilities do not change the same every time you repeat the experiment. For instance:

Flip a coin.
Roll a die and determine whether it is a 5 or not.
Determine whether there was flooding this year in Puerto Vallarta.
Test a randomly selected circuit as it comes off an assembly line and determine whether it is defective or not.
Kick a soccer ball in a random direction and observe whether you score a goal.

We call such an experiment a Bernoulli trial, and refer to the two outcomes -- often arbitrarily -- as success and failure. We will be interested in what happens when we perform a sequence of Bernoulli trials with probability $p$ of success each time. For our random variable $X$ we will count the number of successes. For instance:

Flip a coin 10 times; $X$ = Number of heads.
Roll a die 100 times; $X$ = number of fives you roll.
Provide a property in Puerto Vallarta with flood insurance for 20 years in a zone where flooding occurs every five years on average; $X$ is the number of years, during the 20-year period, during which the property is flooded.
Test 50 randomly selected circuits as they come off an assembly line in which 1% of the circuits are defective; $X$ = Number of defective circuits.

Such a random variable $X$ is called a binomial random variable.

Binomial random variable
A binomial random variable is a random variable that counts the number of successes in a sequence of a fixed number $n$ of Bernoulli trials. It is customary to use $p$ to denote the probability of success in each trial and $q$ to denote the probability of failure (so $q = 1-p$). Examples include the four we have listed above.

Suggested video for this topic:Video by the Khan Academy

%%Examples
Examples include the four we have listed above:

1. \t !3! Flip a coin 10 times; $X$ = Number of heads. \\ \t $\quad n = 10; \ p = .5; \ q = .5$ \gap[40] \t #[The probability of throwing a %%head is .5][La probabilidad de tirar una %%head es .5]# \\ 2. \t !3! Roll a die 100 times; $X$ = number of fives you roll. \\ \t $\quad n = 100; \ p = \dfrac{1}{6}; \ q = \dfrac{5}{6}$ \gap[40] \t #[The probability of rolling a six is $1/6$.][La probabilidad de tirar un seis es $1/6$.]# \\ 3. \t !3! Provide a property in Puerto Vallarta with flood insurance for 20 years in a zone where flooding occurs every five years on average; $X$ is the number of years, during the 20-year period, during which the property is flooded. \\ \t $\quad n = 20; \ p = .2; \ q = .8$ \gap[40] \t #[The probability of flooding in a given year is 1/5.][La probabilidad de inundación en un año determinado es 1/5.]# \\ 4. \t !3! Test 50 randomly selected circuits as they come off an assembly line in which 1% of the circuits are defective; $X$ = Number of defective circuits. \\ \t $\quad n = 50; \ p = .01; \ q = .99$ \gap[40] \t The probability of finding a defective circuit is 1/100.

#[Non example][No ejemplo]#
#[The following is not an example of a binomial random variables:][El siguiente no es un ejemplo de una variable aleatoria binomial:]#

1. \t !3! Select a marble at random from a bag of marbles in which half of them are red, observe whether it is red, and then discard it. Repeat the experiment 10 times; $X$ = Number of red marbles. \\ \t !2! The probability of selecting a red marble changes each time the experiment is performed. For instance, if the first marble is red, then whem you discard it, slightly less than half the remaining marbles are red, so it is slightly less likely to choose a red one the second time.

Some for you

Probability distribution of a binomial random variable

Recall from the %%prevsectut that the probability distribution of a discrete random variable $X$ assigns to each value of $X$ the probability of the event that $X$ equals that value. In particular, if $X$ is a binomial random variable, then $X$ measures the number of successes for a given number $n$ of independent Bernoulli trials, so the possible values of $X$ are $0, 1, 2, ..., n$. Thus, its probability distribution specifies the probabilities that $X = 0$, $X = 1$, ... , $X = n$.

So, calculating the probability distribution of a binomial random variable means calculating the following probabilities:

The probability of 0 successes: \t $P(X = 0)$,\\ The probability of 1 success: \t $P(X=1)$,\\ The probability of 2 successes: \t $P(X=2)$,\\ ... \\ The probability of $n$ successes: \t $P(X=n)$.

The formula below is derived in the textbook. To use it, you should know how to compute the binomial coefficients $C(m,n)$ (See the %%combintut). To assist you, try this Pop-up factorials, permutations and combinations calculator.

Probability distribution of a binomial random variable
If $X$ is the number of successes in a sequence of $n$ independent Bernoulli trials, then

$P(X=x) = C(n,x)p^xq^{n-x} \qquad$ \t

#[where][donde]#
$C(n,x)$ is the number of combinations of $n$ items taken $x$ at a time, sometimes written as $\displaystyle {}_nC_x$ %%or $\binom{n}{x}$. (see the %%combintut). Also,

$n =$ number of trials \\ $p =$ probability of success \\ $q =$ probability of failure $= 1-p$

Suggested video for this topic:Video by Stephanie Glen

%%Examples
1. What is the probability of getting heads exactly twice if you flip a fair coin 6 times?

\\ $x =$ number of successes $= 2$ \\ $n =$ number of trials $= 6$ \\ $p =$ probability of success $= .5$ \\ $q =$ probability of failure $= 1-p = 1-.5 = .5$

Probability of getting 2 heads $= P(X = 2)$ $= C(6, 2)(.5)^2(.5)^{6-2} = (15)(.25)(.0625) \approx .2344$

2. What is the probability of rolling a 6 three times if you roll a fair die five times?

\\ $x =$ number of successes $= 3$ \\ $n =$ number of trials $= 5$ \\ $p =$ probability of success $= \dfrac{1}{6}$ \\ $q =$ probability of failure $= 1-p = 1-\dfrac{1}{6} = \dfrac{5}{6}$

Probability of rolling a 6 three times $= P(X = 2)$ $\displaystyle = C(5, 3)\left(\dfrac{1}{6}\right)^3\left(\dfrac{5}{6}\right)^{5-3} = (10)\left(\dfrac{1}{216}\right)\left(\dfrac{25}{36}\right) \approx .03215$

One for you

#[Using technology][El uso de la tecnología]#
To compute this and other probabilities using technology, try our Binomial distribution utility. To use to compute, say, $P(X = 2)$, enter it as a range by typing "2" in both slots:

$P(\box{\ \ 2 \ \ } \leq X \leq \box{\ \ 2 \ \ })$.

Tabulating the entire probability distribution

Now that we know how to calculate the individual probabilities $P(X = x)$, we can put them all together to obtain the complete probability distribution table. Here is an example:

Now try some of the exercises in Section 9.2 in Finite Mathematics and Applied Calculus.