2.12 Probability4 credits External

We will cover the general theory and examples for:

Equally likely outcomes

Expected value Probability Tree with

replacement Probability Tree

without replacement

Venn Diagrams Contingency Tables Probability from

contingency table Risk & Relative Risk

Textbooks and Resources

DIM is Dimensions, SKY is the Theta edition with the sky tower on it, GREEN is the very old original theta!

Workbook is the Nulake year 12 Mathematics EAS 2.12 book.

There are also a few examples taken from an old workbook – I think Nulake?

The rest of the examples come from my head

Achievement Achievement with Merit

Achievement with Excellence

Apply probability methods in solving problems.

Apply probability methods, using relational thinking, in solving problems.

Apply probability methods, using extended abstract thinking, in solving problems.

Methods include a selection from those related to: experimental distributions relative frequencies two-way tables probability trees risk and relative risk the normal distribution

Remember me? Eli the

Alien

Equally Likely Outcomes

Green p 272 Ex 26.2 p 274 (Q10 and 11 sb done in Contingency Tables later)

Dim p369 Ex 22.01 p370 SKY: 22.01 Homework: get your homework book!

When outcomes are equally likely, the probability of an event is:

Number of outcomes favourable for the event

total number of outcomesOr:

What you wanteverything you’ve got

A pencil case contains 3 pink, 2 green, 6 orange and 9 purple highlighters.One highlighter is chosen at random from the case. What is the probability that the highlighter chosen is:

a) Green b) pink or orange c) not purple d) blue

a) P(green) = 2/20 or 1/10b) P(pink or orange) = 9/20c) P(not purple) = 20-9=11 so 11/20d) P(blue) = zero! 0 nyet zilch…

Note: OR means ______?

‘How many’ questions or easy expected value

Find the value Find the number in the

group Multiply Always give two answers

if there is a context. 23.4 Dimensions

Q: If the probability of getting twins from a ewe (sheep) is 0.3 and Farmer Mac has 200 Ewes, all pregnant, how many ewes are likely to give birth to twins?

How many= prob x number in the group

= 0.3 x200

= 60 - if this wasn’t a ‘nice’ number, state the mathematical answer (2 dp?) then interpret.

Expected Value from a probability table

From a prob table, these are a stinker unless you know what to do.

A sheep farmer knows that the ewes on his land will produce lambs each year as follows:

Calculate the expected value of lambs produced per ewe (sheep)

Dim:23.04 p 398 Green 26.6 p 292 Sky p 316 22.07

NOTE: “Expected value” in this case is like the ‘average’ number:

1x0.69 + 2x0.18 + 3x0.01 + 0x0.12 = 1.08

Lambs 1 2 3 0Prob 0.69 0.18 0.01 0.12

Probability TREES - Times up the arms!

Dim: p372 22.02 (p374) and without replacement p378 22.03 p 379.

Green p284 Then26.4 p286

Workbook p 17 – 21 – nice homework.

Venn Diagrams – an overviewExample1.

We have 18 students in our class, and 2 students take Chem and Physics and Bio. 3 students take just Physics and Bio. 10 students in the class take Bio. NOBODY just takes Chem – hmmm what can we conclude?! 6 students take Physics.Start by making the empty diagram. Put in the ‘intersections’ – ie 2 students and 3 students.

Now look at the BIO section. It’s tempting to write in ‘10’ but you already have 2 and 3 in your Bio circle. So 10-5 = 5. Try and work out the rest!

22.04 DIM – what is ‘U’ and ‘n’ – how do they work?

P. 308 SKYTower

Venn Diagrams – an overview

Venn Diagrams ctd

Probability tables and contingency tables

workbook p 7-10 GIVEN that (teaching notes)

DIM 23.01 and 23.02 (no green book)

What does GIVEN mean?

Tables – it means that we reduce our focus to (usually) one ROW or one COLUMN of a table.

This helps us find the ‘bottom’ of our fraction.

Given that a randomly selected student studies Biology, what is the probability that they are happy with the course? (row or column?)

Make your own question starting ‘given that the student is happy…’ (row or column?)

  Happy

Not Happy

Total

Biology

8 2 10

Chem 0 2 2

Physics

3 3 6

TOTAL

11 7 18

What does GIVEN mean?

Trees – this word means that we usually start half way into our tree, making calculations much easier

Given that I am talking to a girl, how happy is she with the course? This means that you start where the arrow points and go from there.

RISK and relative risk Relative risk of an event for group A compared to the risk

of the same event for group B is the ratio (fraction) of P(A) to P(B) From the table, find the risk for a driver

not wearing a seatbelt who was BELOW the limit: Find the risk for a driver not wearing a seatbelt who was OVER the limit: Now make those two answers into a ‘top heavy’ fraction.

Ans: 46/13 = 3.538 so you are 3-4 x more likely not to wear a seatbelt if you are over the limit.

Note: if you get an answer of (13/46 = 0.28) less than one, it’s often better to restate the question or reverse your fraction but be careful. In the qun above it should be BELOW (13) divided by ABOVE (46) but 0.28 is too hard to interpret. So we say 3.53 – 3-4 x more likely IF you are OVER the limit. – slightly restating the qun.

  Seat belt

No seatbelt

Below BAC limit (80mg)

87%

13%

Above BAC limit (80mg)

54%

46%

Relative risk – example 2

What is the relative risk of a blu-ray needing repair, compared with a dvd?

BLU – 74/161, DVD – 36/239 Make these into decimals

and divide. Rel risk: 0.4596/0.1506 = 3.05 – it’s THREE times

more likely to go bust! HAH!

Note – be careful of your ‘whole group’ 110 or 161?

Relative Risk – example 3• Delayed AirNZ:

62/232 = 0.2674

• Delayed Qantas: 30/118 = 0.2542

• Rel risk: biggest number on top!

= 1.05

About the same!!Interp 0.05 – 5% more likely????

Relative Risk – example 4 This table shows data for a study of

deaths from lung cancer versus other causes of death in the US. The table also shows whether the person was a smoker or not.

Find

1. The risk of a smoker dying of lung cancer

2. The risk of a non-smoker dying of lung cancer

3. Relative risk of dying of lung cancer for a smoker compared to a non-smoker

4. What does the relative risk mean?

  Smoker

Non-smoke

rLung cancer

397 37

Other causes

6919 4614

Total 7316 4651Answers:1. 0.054262. 0.0079553. 6.285 4. What does this mean?!!

Your turn! Relative risk exercises

23.03 DIM p 394 Drivers killed 2005 – 7 (??)

Workbook pp 22 – 27 (best!!)For relative risk: Sky? Orange?

Page 388, Ex25.03 qus 10, 11, 12, 13

Conditional Probability

If we need to find the probability of an event occurring given that another event has already occurred, then we are dealing with conditional probability.

If A and B are two events, then the conditional probability that A occurs given that B already has is written as P(A | B). We say it thus: probability of A GIVEN B.

A rainy exampleOur BC network is affected by the weather. When it rains, the probability of the network

SLOWING down is 3/10. In dry weather, the probability of a SLOW network

is only 1/20. The probability of rain is 1/5.Find the probability that given the network slows

down, it was a rainy day. (I would say “find the prob that it was a rainy day, given the network slows down”, but I’m exposing you to other ways of stating a problem)

Draw a tree to help your thoughts!

Now use the formula above and interpret

P(rains | slow network)What must we do for

the bottom part of our fraction?

A rainy example

A rainy example

Top: P(rains and SLOW) = 1/5 x 3/10

= 3/50Bottom: P(SLOW)= Rain and slow OR sun and slow

= 1/5 x 3/10 + 4/5 x 1/20= 10/100 = 5/50

Answer: 3/50 ÷ 5/50 = 3/5