# Probability by Basic Two-Way Tables | Two-Way Frequency Tables and Probability

$\begin{array}{|c|c|c|c|} \hline &\text{swimmer} &\text{non-swimmer} &\text{total} \\ \hline \text{Boys} & 11 & 10 & \\ \hline \text{Girls} & 13 & 16 & \\ \hline \text{total} & & \\ \hline \end{array}$

## Question 1

(a)     How many boys are in the class?

$\require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline &\text{swimmer} &\text{non-swimmer} &\text{total} \\ \hline \text{Boys} & \bbox[yellow,3px]{11} & \bbox[yellow,3px]{10} & \bbox[orange,3px]{21} \\ \hline \text{Girls} & 13 & 16 & \\ \hline \text{total} & & & \\ \hline \end{array}$
$\bbox[yellow,3px]{11}+\bbox[yellow,3px]{10} = \bbox[orange,3px]{21}$ boys in the class.

(b)     How many girls are in the class?

$\require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline &\text{swimmer} &\text{non-swimmer} &\text{total} \\ \hline \text{Boys} & 11 & 10 & 21 \\ \hline \text{Girls} & \bbox[yellow,3px]{13} & \bbox[yellow,3px]{16} & \bbox[orange,3px]{29} \\ \hline \text{total} & & & \\ \hline \end{array}$
$\bbox[yellow,3px]{13}+\bbox[yellow,3px]{16} = \bbox[orange,3px]{29}$ girls in the class.

(c)     How many students who can swim are in the class?

$\require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline &\text{swimmer} &\text{non-swimmer} &\text{total} \\ \hline \text{Boys} & \bbox[yellow,3px]{11} & 10 & 21 \\ \hline \text{Girls} & \bbox[yellow,3px]{13} & 16 & 29 \\ \hline \text{total} & \bbox[orange,3px]{24} & & \\ \hline \end{array}$
$\bbox[yellow,3px]{11}+\bbox[yellow,3px]{13} = \bbox[orange,3px]{24}$ swimmers in the class.

(d)     How many students cannot swim in the class?

$\require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline &\text{swimmer} &\text{non-swimmer} &\text{total} \\ \hline \text{Boys} & 11 & \bbox[yellow,3px]{10} & 21 \\ \hline \text{Girls} & 13 & \bbox[yellow,3px]{16} & 29 \\ \hline \text{total} & 24 & \bbox[orange,3px]{26} & \\ \hline \end{array}$
$\bbox[yellow,3px]{10}+\bbox[yellow,3px]{16} = \bbox[orange,3px]{26}$ non-swimmers in the class.

(e)     How many students are in the class?

$\require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline &\text{swimmer} &\text{non-swimmer} &\text{total} \\ \hline \text{Boys} & 11 & 10 & \bbox[yellow,3px]{21} \\ \hline \text{Girls} & 13 & 16 & \bbox[yellow,3px]{29} \\ \hline \text{total} & \bbox[yellow,3px]{24} & \bbox[yellow,3px]{26} & \bbox[orange,3px]{50} \\ \hline \end{array}$
\begin{align} \bbox[yellow,3px]{24}+\bbox[yellow,3px]{26} &= \bbox[orange,3px]{50} \\ \bbox[yellow,3px]{21}+\bbox[yellow,3px]{29} &= \bbox[orange,3px]{50} \end{align} students in the class.

(f)     How many students are boys or swimmers?

$\require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline &\text{swimmer} &\text{non-swimmer} &\text{total} \\ \hline \text{Boys} & \bbox[yellow,3px]{11} & \bbox[yellow,3px]{10} & 21 \\ \hline \text{Girls} & \bbox[yellow,3px]{13} & 16 & 29 \\ \hline \text{total} & 24 & 26 & 50 \\ \hline \end{array}$
$\bbox[yellow,3px]{11}+\bbox[yellow,3px]{10}+\bbox[yellow,3px]{13} = \bbox[orange,3px]{34}$

(g)     How many students are boys and swimmers?

$\require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline &\text{swimmer} &\text{non-swimmer} &\text{total} \\ \hline \text{Boys} & \bbox[orange,3px]{11} & 10 & 21 \\ \hline \text{Girls} & 13 & 16 & 29 \\ \hline \text{total} & 24 & 26 & 50 \\ \hline \end{array}$
$\bbox[orange,3px]{11}$

(h)     Find the probability that a randomly chosen student is a girl.

$\require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline &\text{swimmer} &\text{non-swimmer} &\text{total} \\ \hline \text{Boys} & 11 & 10 & 21 \\ \hline \text{Girls} & 13 & 16 & \bbox[orange,3px]{29} \\ \hline \text{total} & 24 & 26 & \bbox[orange,3px]{50} \\ \hline \end{array}$
$\displaystyle \frac{\bbox[orange,3px]{29}}{\bbox[orange,3px]{50}}$

(i)     Find the probability that a randomly chosen student is a non-swimmer.

$\require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline &\text{swimmer} &\text{non-swimmer} &\text{total} \\ \hline \text{Boys} & 11 & 10 & 21 \\ \hline \text{Girls} & 13 & 16 & 29 \\ \hline \text{total} & 24 & \bbox[orange,3px]{26} & \bbox[orange,3px]{50} \\ \hline \end{array}$
$\displaystyle \frac{\bbox[orange,3px]{26}}{\bbox[orange,3px]{50}} = \frac{\bbox[orange,3px]{13}}{\bbox[orange,3px]{25}}$

(j)     Find the probability that a randomly chosen student is a girl who can swim.

$\require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline &\text{swimmer} &\text{non-swimmer} &\text{total} \\ \hline \text{Boys} & 11 & 10 & 21 \\ \hline \text{Girls} & \bbox[orange,3px]{13} & 16 & 29 \\ \hline \text{total} & 24 & 26 & \bbox[orange,3px]{50} \\ \hline \end{array}$
$\displaystyle \frac{\bbox[orange,3px]{13}}{\bbox[orange,3px]{50}}$

## Question 2

$\begin{array}{|c|c|c|c|} \hline &\text{Heavy} &\text{Light} &\text{total} \\ \hline \text{Boys} & & 21 & 33 \\ \hline \text{Girls} & 29 & & \\ \hline \text{total} & & & 80 \\ \hline \end{array}$

(a)     How many heavy boys are there?

$\begin{array}{|c|c|c|c|} \hline &\text{Heavy} &\text{Light} &\text{total} \\ \hline \text{Boys} & \bbox[orange,3px]{12} & \bbox[yellow,3px]{21} & \bbox[yellow,3px]{33} \\ \hline \text{Girls} & 29 & & \\ \hline \text{total} & & & 80 \\ \hline \end{array}$
$\displaystyle \bbox[yellow,3px]{33}-\bbox[yellow,3px]{21}=\bbox[orange,3px]{12}$

(b)     How many heavy students are there?

$\begin{array}{|c|c|c|c|} \hline &\text{Heavy} &\text{Light} &\text{total} \\ \hline \text{Boys} & \bbox[yellow,3px]{12} & 21 & 33 \\ \hline \text{Girls} & \bbox[yellow,3px]{29} & & \\ \hline \text{total} & \bbox[orange,3px]{41} & & 80 \\ \hline \end{array}$
$\displaystyle \bbox[yellow,3px]{12}+\bbox[yellow,3px]{29}=\bbox[orange,3px]{41}$

(c)     How many light students are there?

$\begin{array}{|c|c|c|c|} \hline &\text{Heavy} &\text{Light} &\text{total} \\ \hline \text{Boys} & 12 & 21 & 33 \\ \hline \text{Girls} & 29 & & \\ \hline \text{total} & \bbox[yellow,3px]{41} & \bbox[orange,3px]{39} & \bbox[yellow,3px]{80} \\ \hline \end{array}$
$\displaystyle \bbox[yellow,3px]{80}-\bbox[yellow,3px]{41}=\bbox[orange,3px]{39}$

(d)     How many light girls are there?

$\begin{array}{|c|c|c|c|} \hline &\text{Heavy} &\text{Light} &\text{total} \\ \hline \text{Boys} & 12 & \bbox[yellow,3px]{21} & 33 \\ \hline \text{Girls} & 29 & \bbox[orange,3px]{18} & \\ \hline \text{total} & 41 & \bbox[yellow,3px]{39} & 80 \\ \hline \end{array}$
$\displaystyle \bbox[yellow,3px]{39}-\bbox[yellow,3px]{21}=\bbox[orange,3px]{18}$

(e)     How many girls are there?

$\begin{array}{|c|c|c|c|} \hline &\text{Heavy} &\text{Light} &\text{total} \\ \hline \text{Boys} & 12 & 21 & 33 \\ \hline \text{Girls} & \bbox[yellow,3px]{29} & \bbox[yellow,3px]{18} & \bbox[orange,3px]{47} \\ \hline \text{total} & 41 & 39 & 80 \\ \hline \end{array}$
$\displaystyle \bbox[yellow,3px]{29}+\bbox[yellow,3px]{18}=\bbox[orange,3px]{47}$

(f)     Find the probability that a randomly chosen student is a girl.

$\begin{array}{|c|c|c|c|} \hline &\text{Heavy} &\text{Light} &\text{total} \\ \hline \text{Boys} & 12 & 21 & 33 \\ \hline \text{Girls} & 29 & 18 & \bbox[orange,3px]{47} \\ \hline \text{total} & 41 & 39 & \bbox[orange,3px]{80} \\ \hline \end{array}$
$\displaystyle \frac{\bbox[orange,3px]{47}}{\bbox[orange,3px]{80}}$

(g)     Find the probability that a randomly chosen student is a light boy.

$\begin{array}{|c|c|c|c|} \hline &\text{Heavy} &\text{Light} &\text{total} \\ \hline \text{Boys} & 12 & \bbox[orange,3px]{21} & 33 \\ \hline \text{Girls} & 29 & 18 & 47 \\ \hline \text{total} & 41 & 39 & \bbox[orange,3px]{80} \\ \hline \end{array}$
$\displaystyle \frac{\bbox[orange,3px]{21}}{\bbox[orange,3px]{80}}$

(h)     Find the probability that a randomly chosen student is a girl or a heavy student.

$\begin{array}{|c|c|c|c|} \hline &\text{Heavy} &\text{Light} &\text{total} \\ \hline \text{Boys} & \bbox[yellow,3px]{12} & 21 & 33 \\ \hline \text{Girls} & \bbox[yellow,3px]{29} & \bbox[yellow,3px]{18} & 47 \\ \hline \text{total} & 41 & 39 & \bbox[orange,3px]{80} \\ \hline \end{array}$
$\displaystyle \frac{\bbox[yellow,3px]{12}+\bbox[yellow,3px]{29} +\bbox[yellow,3px]{18}}{\bbox[orange,3px]{80}} = \frac{\bbox[yellow,3px]{59}}{\bbox[orange,3px]{80}}$