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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 $28$??
$$ \left(\begin{matrix} 28 \\ 3 \end{matrix}\right) $$
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??
$$ \left(\begin{matrix} 20 \\ 1 \end{matrix}\right) $$
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??
$$ 0.1 \times 0.1 \times 0.9 $$
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??
$$ \left(\begin{matrix} 3 \\ 2 \end{matrix}\right) $$
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??
$$ \left(\begin{matrix} 3 \\ 2 \end{matrix}\right) \times 0.1 \times 0.1 \times 0.9 $$
$$X ~ B(n, p)$$ 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 $X ~ B(n, p)$??
$$ \left( \begin{matrix}n \\ r\end{matrix} \right) p^r(1-p)^{n-r} $$
$$P(X = r) = \left( \begin{matrix}n \\ r\end{matrix} \right) p^r(1-p)^{n-r}$$ 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.
$$P(X = r) = \left( \begin{matrix}n \\ r\end{matrix} \right) p^r(1-p)^{n-r}$$ 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 $X ~ B(n, p)$??
$$ (1 - p)^n $$
$$P(X = 0)$$ 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 $X ~ B(n, p)$??
$$ p^n $$
$$P(X = n)$$ 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 $p$.
2021-02-09
$$P(X \le 1)$$ What’s the long-winded way of working this out??
$$ P(X = 0) + P(X = 1) $$
$$P(X \le 6)$$ What’s the long-winded way of working this out??
$$ P(X = 0) + P(X = 1) + P(X = 2) + … $$
$$P(X > 5)$$ How could you rewrite this so it can be used with a binomial cumulative distribution??
$$ 1 - P(X \le 5) $$
$$P(X \ge 7)$$ How could you rewrite this so it can be used with a binomial cumulative distribution??
$$ 1 - P(X \le 6) $$
$$P(X < 10)$$ How could you rewrite this so it can be used with a binomial cumulative distribution??
$$ P(X \le 9) $$
$$P(X > 16)$$ How could you rewrite this so it can be used with a binomial cumulative distribution??
$$ 1 - P(X \le 16) $$
$$P(6 < X \le 10)$$ How could you rewrite this so it can be used with a binomial cumulative distribution??
$$ P(X \le 10) - P(X \le 6) $$
$$P(\text{“at most 8”})$$ How could you rewrite this so it can be used with a binomial cumulative distribution??
$$ P(X \le 8) $$
$$P(\text{“no more than 3”})$$ How could you rewrite this so it can be used with a binomial cumulative distribution??
$$ P(X \le 3) $$
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Metadata
date: 2021-02-04 09:54
tags:
- '@?stats'
- '@?probability'
- '@?distributions'
- '@?year-1'
- '@?school'
- '@?public'
title: Stats - Binomial Distribution