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(14.2) Binomial Distribution

Rolling 3 dice

Consider an experiement where you roll 3 dice. Let $X$ be the number of ones rolled. alt-text

$X=0$
If zero 1s are rolled, not 1 must be rolled on all three dice. alt-text
The probability of rolling [not 1]-[not 1]-[not 1] is $\biggl($
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$\biggl)\biggl($
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$\biggl)\biggl($
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$\biggl)$

The probability of rolling zero 1s, $P\left(X=0\right)=\biggl($
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$\biggl)$

$X=1$
One 1 can be rolled in the following way. alt-text
The probability of rolling [1]-[not 1]-[not 1] is $\Bigl($ $\Bigl)\Bigl($ $\Bigl)\Bigl($ $\Bigl)=\Bigl($ $\Bigl)\Bigl($ $\Bigl)$

However, one 1 can also be rolled like this. alt-text
The probability of rolling [not 1]-[1]-[not 1] is $\Bigl($ $\Bigl)\Bigl($ $\Bigl)\Bigl($ $\Bigl)=\Bigl($ $\Bigl)\Bigl($ $\Bigl)$

Finally, one 1 can also be rolled like this. alt-text
The probability of rolling [not 1]-[not 1]-[1] is $\Bigl($ $\Bigl)\Bigl($ $\Bigl)\Bigl($ $\Bigl)=\Bigl($ $\Bigl)\Bigl($ $\Bigl)$

The probability of rolling one 1, $P\left(X=1\right)=$ $\Bigl($ $\Bigl)\Bigl($ $\Bigl)$

$X=2$
Two 1s can be rolled in the following three ways. alt-text or alt-text or alt-text
The probability for any one of these ways is $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$

The probability of rolling two 1s, $P\left(X=2\right)=$ $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$

$X=3$
Three 1s can only be rolled in the following way. alt-text

The probability of rolling three 1s, $P\left(X=3\right)=\Bigl($ $\Bigl)$

Complete the probability distribution table for rolling 3 dice where $X$ is the number of 1s rolled.
$x$ $0$ $1$ $2$ $3$
$P\left(X=x\right)$ $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$

Rolling 4 dice

Consider an experiement where you roll 4 dice. Let $X$ be the number of ones rolled. alt-text

$X=0$
The probability of rolling zero 1s, $P\left(X=0\right)=\Bigl($ $\Bigl)$

$X=1$
One 1 can be rolled in the following way. alt-text
Let's record this as xoxx where x represents not 1 and o represents 1.
List the other possible ways of rolling one 1. Separate your answers with commas.
The probability for any one of these ways is $\Bigl($ $\Bigl)\Bigl($ $\Bigl)$

The probability of rolling one 1, $P\left(X=1\right)=$ $\Bigl($ $\Bigl)\Bigl($ $\Bigl)$

$X=2$
Two 1s can be rolled in the following way. alt-text
Let's record this as ooxx where x represents not 1 and o represents 1.
List the other possible ways of rolling two 1s. Separate your answers with commas.
The probability for any one of these ways is $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$

The probability of rolling two 1s, $P\left(X=2\right)=$ $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$

$X=3$
List all the possible ways of rolling three 1s using x and o. Separate your answers with commas.alt-text
The probability for any one of these ways is $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$

The probability of rolling three 1s, $P\left(X=3\right)=$ $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$

$X=4$
The probability of rolling four 1s, $P\left(X=4\right)=\Bigl($ $\Bigl)$

Complete the probability distribution table for rolling 4 dice where $X$ is the number of 1s rolled.
$x$ $0$ $1$ $2$ $3$ $4$
$P\left(X=x\right)$ $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$ $\Bigl($ $\Bigl)$

Do you see patterns similar to those found in binomial expansions?
$x$ $0$ $1$ $2$ $3$ $4$
$P\left(X=x\right)$ $\color{silver}{\binom{4}{0}\left(\frac{1}{6}\right)^0} \left(\frac{5}{6}\right)^4$ $\binom{4}{1}\left(\frac{1}{6}\right)^\color{silver}{1} \left(\frac{5}{6}\right)^3$ $\binom{4}{2}\left(\frac{1}{6}\right)^2 \left(\frac{5}{6}\right)^2$ $\binom{4}{3}\left(\frac{1}{6}\right)^3 \left(\frac{5}{6}\right)^\color{silver}{1}$ $\color{silver}{\binom{4}{4}}\left(\frac{1}{6}\right)^4 \color{silver}{\left(\frac{5}{6}\right)^0}$

These are examples of binomial distributions. For binomial distributions:
  • there is a set number of trials, $n$. In the above example, $n=$
  • the outcome of each trial is either a success or not a success
  • trials are independent of each other
  • there is a probability for success, $p$. In our example, it was the probability of rolling a 1. $p=$
Binomial distributions can be represented as $X\sim B\left(n,p\right)$ where $\sim$ means "is distributed as".
In the above example, $X\sim B\bigl($ $\bigl)$

Example

Consider an experiment where you roll 5 dice. What is the probability of rolling exactly three 1s?
$X\sim B\bigl($ $\bigl)\qquad P\left(X=3\right)=$ $\left(\frac{1}{6}\right)$ $\left(\frac{5}{6}\right)$ $=$
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*Write your final answer in simplest form

Example

In a city, the probability of rain on any given day is 30%. For a given week (7 days), find the following probabilities and fill in the probability distribution table:
  • it doesn't rain
  • it rains on at least 1 day

      • $x$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$
      $P\left(X=x\right)$