Simple Probability Diagrams Flashcards

All 19 cards in this deck

What is a two-way table?

A table that shows the frequencies for two different categories at the same time, with row totals, column totals and a grand total.

What are the steps to complete a two-way table?

e.g. 20 people: 12 are adults, 8 are children; 5 of the adults chose tea, 9 people chose tea altogether

1. Write in the values given: adults–tea $= 5$.
2. Subtract from a row or column total to fill a gap: adults–coffee $= 12 - 5 = 7$.
3. Use the other totals for the rest, then check every row and column adds up: children–tea $= 9 - 5 = 4$.

One person is chosen at random from everyone in a two-way table. How do you find the probability that they are in a particular cell?

Divide that cell's frequency by the grand total.

e.g. cell of $7$ out of $20$ people gives $\frac{7}{20}$.

True or false? In a completed two-way table, the row totals add up to the same number as the column totals.

True. Both add up to the grand total.

True or false? The numbers inside a two-way table are probabilities, so they should add up to $1$.

False. They are frequencies, and they add up to the grand total.

What is a frequency tree?

A diagram whose branches split a total into groups, with the frequency (a whole number) written at the end of each branch.

True or false? The numbers written on a frequency tree are probabilities.

False. They are frequencies — whole numbers of items or people.

In a frequency tree, what must be true of the frequencies on the branches coming out of one point?

They add up to the frequency at that point.

e.g. a group of $60$ splitting into $15$ and $45$.

What are the steps to complete a frequency tree?

e.g. $100$ people take a test: $60$ are men, $15$ of the men passed, $40$ people passed altogether

1. Write the total and the first split: $60$ men, $100 - 60 = 40$ women.
2. Complete a split using its own group total: men who failed $= 60 - 15 = 45$.
3. Use the overall totals for the rest: women who passed $= 40 - 15 = 25$, women who failed $= 40 - 25 = 15$.

How do you find a probability from a completed frequency tree?

e.g. P(a person chosen at random passed the test)

Add the end frequencies for that outcome and divide by the total at the start of the tree.

e.g. $\frac{15 + 25}{100} = \frac{40}{100}$.

What does $A \cap B$ mean?

The intersection: the elements that are in both $A$ and $B$ — the overlap of the two circles.

True or false? $A \cup B$ means the elements that are in both $A$ and $B$.

False. That is $A \cap B$; the union $A \cup B$ is everything in $A$, in $B$, or in both.

What does $A'$ mean?

The complement of $A$: all the elements of the universal set that are not in $A$.

In a Venn diagram, what does $\xi$ (also written $E$) stand for?

The universal set: all the elements being considered, shown by the rectangle around the circles.

A number is chosen at random from the universal set $\xi$. How do you find $P(A \cup B)$ from a completed Venn diagram?

Count the elements in every region inside either circle and divide by the total number of elements in $\xi$.

e.g. $5$ elements in $A \cup B$ out of $9$ gives $\frac{5}{9}$.

What are the steps to complete a two-set Venn diagram from given frequencies?

e.g. $30$ people: $18$ like tea, $12$ like coffee, $7$ like both

1. Write the intersection first: $7$ in tea $\cap$ coffee.
2. Subtract it from each set total: tea only $18-7=11$, coffee only $12-7=5$.
3. Subtract all three regions from the universal total for outside: $30-(11+7+5)=7$.

True or false? If $18$ people like tea and $7$ like both tea and coffee, you write $18$ in the 'tea only' region of the Venn diagram.

False. $18$ is the total for the whole tea circle; 'tea only' is $18-7=11$.

Which region of a three-set Venn diagram should you fill in first, and why?

The middle, $A \cap B \cap C$ — every other overlap includes it, so it must be subtracted out of them.

What are the steps to complete a three-set Venn diagram from overlap totals?

e.g. $30$ people; $4$ do all three; $10$ do $A$ and $B$; $7$ do $B$ and $C$; $6$ do $A$ and $C$; $15$ do $A$; $18$ do $B$; $13$ do $C$

1. Put the 'all three' number in the middle: $4$.
2. Pair region $=$ pair total $-$ middle: $A\cap B$ only $=6$, $B\cap C$ only $=3$, $A\cap C$ only $=2$.
3. 'Only' region $=$ set total $-$ its other regions: $A$ only $=3$, $B$ only $=5$, $C$ only $=4$; outside $=30-27=3$.