BIOR92 Maths Practice

This is a short review exercise to refresh your memory of some basic math skills you will need in order to understand the material covered in this course. I strongly encourage you to complete the exercises before the term starts.

1. Proportions

There are 201 individuals of Drosophila melanogaster in a vial. 105 flies have white eyes, while the remainder have red eyes. What proportion of the flies in the vial have white eyes?

2. Proportions

You have 5 vials, each containing 166 D. melanogaster flies. The vials contain 45, 36, 38, 10, 3 flies with white eyes. What is the average proportion of white eyed flies per vial?

3. Weighted Mean

The table below presents fly count data for three different vials. What is the average proportion of white eyed flies per vial? (Hint: you must calculate a weighted mean).

Vial	Total_Flies	White_Eyes
1	162	22
2	69	15
3	66	2

4. Mean & variance

The formula for a sample mean is: $\overline{x} = \frac{\sum_i^n (x_i)}{n}$

The formula for a sample variance is: $s^2 = \frac{\sum_i^n (x_i - \bar{x})^2}{n - 1}$

Calculate the mean and variance for the following data by hand: 80, 85, 65, 56, 24, 46, 14, 79

5. Logarithms

Simplify the following:

$\ln ( x \times y)$
$\ln \left( \frac{x}{y} \right)$
$\ln \left( \frac{1}{x} \right)$
$\ln \left( 1 \right)$
$\ln \left( x^n \right)$

6. Factor

$x^2 + 2xy + y^2$
$2x^2 - 3xy + y^2$
$2x^2 - 4 x y + 2y^2$

7. Solve for x:

$a x + (1 - x)(1 - a) = 0$
$a = \frac{2x}{b}$
$a = \frac{2x^2}{b}$
$y = (1 - a)^{(x/2)}$

8. Relative fitness

The average number of offspring for three populations of an insect are $w_1 =$ 35, $w_2=$ 80, and $w_3 =$ 182. Calculate the offspring numbers relative to $w_1$.

9. Probability

You roll a 16-sided dice 5 times. What is the probability that you roll a 2 every time?

What is the probability that you roll a 2 zero times?

10. Probability

You reach into your very large and messy sock drawer - which contains 0.515 Blue and 0.485 Gold socks (unpaired) - and pull out 2 to wear today. What is the probability that you will be wearing

Matching Blue socks?
Matching Gold socks?
One Blue and one Gold sock?

Your inexplicably sockless friend asks to borrow a pair of socks. Being a generous soul, you reach into your sock drawer and pull them out a pair of socks too. What is the probability that:

You will both be wearing matching Blue socks?
You will be wearing matching socks of a different color?
You will both be wearing 1 Blue and 1 Gold sock?

--- title: "BIOR92 Maths Practice" --- This is a short review exercise to refresh your memory of some basic math skills you will need in order to understand the material covered in this course. I strongly encourage you to complete the exercises _before_ the term starts. ## 1. Proportions ```{r} #| echo: FALSE set.seed(123456) ntot <- sample(c(160:204), size=1) nwhite <- sample(c(64:125), size=1) ``` There are `r ntot` individuals of *Drosophila melanogaster* in a vial. `r nwhite` flies have white eyes, while the remainder have red eyes. What proportion of the flies in the vial have white eyes? ```{r} #| echo: FALSE #nwhite/ntot ``` \vspace{1.5in} ## 2. Proportions ```{r} #| echo: FALSE nTot <- sample(c(160:204), size=1) nWhites <- sample(c(1:50), size=5) ``` You have 5 vials, each containing `r nTot` *D. melanogaster* flies. The vials contain `r nWhites` flies with white eyes. What is the average proportion of white eyed flies per vial? ```{r} #| echo: FALSE #mean(nWhites/nTot) ``` \newpage ## 3. Weighted Mean The table below presents fly count data for three different vials. What is the average proportion of white eyed flies per vial? (**Hint:** you must calculate a weighted mean). ```{r} #| echo: FALSE library(knitr) Vial <- c("1", "2", "3") Total_Flies <- c(sample(c(160:204), size=1), sample(c(40:80), size=1), sample(c(60:70), size=1)) White_Eyes <- c(sample(c(20:46), size=1), sample(c(10:30), size=1), sample(c(1:20), size=1)) kable(cbind(Vial,Total_Flies,White_Eyes)) ``` ```{r} #| echo: FALSE #weights <- Total_Flies/sum(Total_Flies) #pWhites <- White_Eyes/Total_Flies #weightedMean <- sum(weights*pWhites) #weights #pWhites #weightedMean ``` \vspace{2in} ## 4. Mean & variance The formula for a sample mean is: $\overline{x} = \frac{\sum_i^n (x_i)}{n}$ The formula for a sample variance is: $s^2 = \frac{\sum_i^n (x_i - \bar{x})^2}{n - 1}$ ```{r} #| echo: FALSE samp <- sample(c(1:100), size=8) ``` Calculate the mean and variance for the following data by hand: `r samp` ```{r} #| echo: FALSE #u <- mean(samp) #v <- var(samp) #u #v ``` \vspace{2in} ## 5. Logarithms Simplify the following: a. $\ln ( x \times y)$ b. $\ln \left( \frac{x}{y} \right)$ c. $\ln \left( \frac{1}{x} \right)$ d. $\ln \left( 1 \right)$ e. $\ln \left( x^n \right)$ \vspace{2in} ## 6. Factor a. $x^2 + 2xy + y^2$ b. $2x^2 - 3xy + y^2$ c. $2x^2 - 4 x y + 2y^2$ \vspace{2in} ## 7. Solve for x: a. $a x + (1 - x)(1 - a) = 0$ b. $a = \frac{2x}{b}$ c. $a = \frac{2x^2}{b}$ d. $y = (1 - a)^{(x/2)}$ \vspace{2in} ## 8. Relative fitness ```{r} #| echo: FALSE v1 = sample(c(1:200), size=1) v2 = sample(c(1:200), size=1) v3 = sample(c(1:200), size=1) ``` The average number of offspring for three populations of an insect are $w_1 =$ `r v1`, $w_2=$ `r v2`, and $w_3 =$ `r v3`. Calculate the offspring numbers relative to $w_1$. ```{r} #| echo: FALSE #c(v1/v1, v2/v1, v3/v1) ``` \vspace{2in} ## 9. Probability ```{r} #| echo: FALSE nSide <- sample(c(10:20), size=1) n <- sample(c(3:7), size=1) side <- 50 while(side > nSide){ side = sample(c(1:20), size=1) } ``` You roll a `r nSide`-sided dice `r n` times. What is the probability that you roll a `r side` every time? ```{r} #| echo: FALSE #(1/nSide)^n ``` What is the probability that you roll a `r side` zero times? ```{r} #| echo: FALSE #(1 - (1/nSide))^n ``` \vspace{2in} ## 10. Probability ```{r} #| echo: FALSE pBlue = round(runif(1), digits=3) pGold = 1 - pBlue ``` You reach into your very large and messy sock drawer - which contains `r pBlue` Blue and `r pGold` Gold socks (unpaired) - and pull out 2 to wear today. What is the probability that you will be wearing - Matching Blue socks? - Matching Gold socks? - One Blue and one Gold sock? ```{r} #| echo: FALSE # pMatchBlue <- pBlue^2 # pMatchGold <- (1 - pBlue)^2 # pMisMatch <- 2*pBlue*(1 - pBlue) # pMatchBlue # pMatchGold # pMisMatch ``` Your inexplicably sockless friend asks to borrow a pair of socks. Being a generous soul, you reach into your sock drawer and pull them out a pair of socks too. What is the probability that: - You will both be wearing matching Blue socks? - You will be wearing matching socks of a different color? - You will both be wearing 1 Blue and 1 Gold sock? ```{r} #| echo: FALSE # pBothMatchBlue <- pMatchBlue^2 # pMatchDifferent <- 2*pMatchBlue*pMatchGold # pBothMisMatch <- pMisMatch^2 # pBothMatchBlue # pMatchDifferent # pBothMisMatch ```