# Stats - Binomial Distribution > Source: https://ollybritton.com/notes/a-level/maths/topics/stats/binomial-distribution/ · Updated: 2021-02-04 · Tags: stats, probability, distributions, year-1, school ## See Also - [Stats - Statistical Distributions](https://ollybritton.com/notes/a-level/maths/topics/stats/statistical-distributions/) - [Stats - Probability](https://ollybritton.com/notes/a-level/maths/topics/stats/probability/) - [redacted](https://ollybritton.com/404) ## 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?? 1. There are a fixed number of trials 2. There are two possible outcomes 3. There is a fixed probability of success 4. 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?? $$ P(X \le 3) $$ --- Olly Britton — https://ollybritton.com. 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