Stats - Binomial Distribution
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Flashcards
What $ \left(\begin{matrix} n \ r \end{matrix}\right) $ could you write for the number of ways you can pick $3$ students from a class of $28$?
What $ \left(\begin{matrix} n \ r \end{matrix}\right) $ could you write for the number of ways you could pick $2$ left-handed people from a total of $3$ people?
\[\left(\begin{matrix} 3 \\\\ 2 \end{matrix}\right)\]What $ \left(\begin{matrix} n \ r \end{matrix}\right) $ could you write for the number of ways you could pick $1$ defective screw out of $20$ defective screws?
In a group of three people, two are left-handed and one is not. The probability of being left-handed is $0.1$, so the probability of being right-handed is $0.9$. What is the probability for each permutation?
In a group of three people, two are left-handed and one is not. The probability of being left-handed is $0.1$, so the probability of being right-handed is $0.9$. What $ \left(\begin{matrix} n \ r \end{matrix}\right) $ could you write for the number of outcomes?
In a group of three people, two are left-handed and one is not. The probability of being left-handed is $0.1$, so the probability of being right-handed is $0.9$. There are $ \left(\begin{matrix} 3 \ 2 \end{matrix}\right) $ possible outcomes and the probability of each one is $0.1 \times 0.1 \times 0.9$. How could you write for the overall probability?
\[X ~ B(n, p)\]
What does this mean?
$X$ has a binomial distribution with $n$ trials and probability of success $p$.
What are the 4 criteria for $X$ being modelled with a binomial distribution?
- There are a fixed number of trials
- There are two possible outcomes
- There is a fixed probability of success
- The trials are independent of each other
What is the formula for $P(X = r)$ if $X ~ B(n, p)$?
\[P(X = r) = \left( \begin{matrix}n \\\\ r\end{matrix} \right) p^r(1-p)^{n-r}\]
What does $r$ represent?
The number of successes.
\[P(X = r) = \left( \begin{matrix}n \\\\ r\end{matrix} \right) p^r(1-p)^{n-r}\]
What does $n$ represent?
The number of trials.
\[P(X = r) = \left( \begin{matrix}n \\\\ r\end{matrix} \right) p^r(1-p)^{n-r}\]
What does $p$ represent?
The probability of success.
\[P(X = r) = \left( \begin{matrix}n \\\\ r\end{matrix} \right) p^r(1-p)^{n-r}\]
If $p^r$ is the probability of $r$ successes and $(1-p)^{n-r}$ is the probability of the number of failures, can you explain $p^r(1 - p)^{n - r}?
It’s the probability that each outcome is true.
\[P(X = r) = \left( \begin{matrix}n \\\\ r\end{matrix} \right) p^r(1-p)^{n-r}\]
If $p^r$ is the probability of $r$ successes, can you explain $n - r$?
If there are $r$ successes out of $n$, then $n - r$ must be the number of failures.
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\[P(X = r) = \left( \begin{matrix}n \\\\ r\end{matrix} \right) p^r(1-p)^{n-r}\]If $p$ is the probability of success, can you explain $(1 - p)$? In a binomial distribution there are only two outcomes, so $(1 - p)$ is the probability of failure.
Why can’t use you use the binomial distribution to model an experiment where you have red counters and green counters in a bag and you wish to find the probability that removing $5$ counters would contain $3$ greens?
Because the trials aren’t independent of each other.
\[P(X = 0)\]
What is this equal to if $X ~ B(n, p)$?
\[P(X = 0)\]
If $X ~ B(n, p)$, what does this mean in simple terms?
The probability of no successes.
\[P(X = n)\]
What is this equal to if $X ~ B(n, p)$?
\[P(X = n)\]
If $X ~ B(n, p)$, what does this mean in simple terms?
The probability of no failures.
On a Classwiz calculator, how can you work out $P(X = r)$ if $X ~ B(n, p)$?
- Distribution
- Binomial PD
- Variable
- Enter values for $x$, $n$ and $p$.
On a Classwiz calculator, how can you work out $P(X \le r)$ if $X ~ B(n, p)$?
- Distribution
- Down, Binomial CD
- Variable
- Enter values for $x$, $n$ and $p$.