Independent Events

Slideshow 54, Mathematics

Mr Richard Sasaki, Room 307

Objectives

• Recalling the meaning of “with replacement” and “without replacement”

• Understand independence and calculating probabilities about 2 events

Vocabulary

We need a bit of a review.

Event (Trial) -

The thing that is taking place (eg: Rolling a die)

Value - Possible outcomes for the event (for a die: 1, 2, 3, 4, 5, 6)

Frequency - The number of times a value appears in an experiment.

Independence

In the Winter Homework, Independence was mentioned. What is it again?

Independence for events is where one event doesn’t affect another.

This means that no matter what happens in one event, the probabilities for the other event are exactly the same.

Last lesson we looked at pulling objects out of a bag and “with replacement” and “without replacement”. Let’s review those meanings.

With and Without Replacement

With Replacement –

After an event occurs, everything is “reset” (put back as it was) so when we repeat, nothing has changed.

Without Replacement –

After an event occurs, whatever happened is removed from the event, causing all future occurrences to have differing probabilities.

Which of these shows independence?

With Replacement

# of outcomes

Flipping a Coin…Twice!

Let’s consider flipping an unbiased coin twice. What are the possibilities we can get?Let’s list them…①　Heads, Heads

②　Heads, Tails③　 Tails, Heads④　Tails, Tails

Note – Heads, Tails is different to Tails, Heads. When listing possible outcomes, order does have meaning.

Probability (of happening)P(H, H) = P(H, T) = P(T, H) = P(T, T) =

Why is each ¼?# of successes =

There are 4 combinations.

Flipping a Coin…Twice!

How about the following?

P(A Heads and a Tails) =24

Order isn’t mentioned.P(H and T) = P(H, T) + P(T, H).

P(A Heads or a Tails) =1We always get heads or

tails!The terms “and”, “or” and “,” are all very different when there are 2 or more events taking place.

And - Both must happen (any order)Or - At least one of them must happen

, - Both must happen in the given order

Answers - Easy

P(Tails) = ½

Yes, their outcomes don’t affect each other.

P(H, T) = ¼ P(At least one heads) =

P(Exactly one heads) =

P(No tails) =

No. Both events are independent with the same probabilities for each outcome.

1, 2, 3P(2) =

1, 1 2, 1 3, 11, 2 2, 2 3, 21, 3 2, 3 3, 3

P(1, 3) =

P(2, 2) =

P(No 4) =

Answers - Hard

16

P(2, 4) =

P(A 1 and a 3) = P(Exactly one 2) =

P(Total of 7) = P(Total of less than 3) =

P(A 3 and a 4) = P(Total of 7) =

P(Both Even) = P(Both less than or equal to 4) =

12

P(H, 4) =

P(Tails and greater than 2) =

P(Two sixes) = 0

Because order is considered. The coin is flipped first so we can’t flip a 3.

An Introduction to Permutations

Many of our examples involved choosing 2 from a group of some number of options with repetition allowed (picking the same number twice).

Two options, two picked (Coin) - 4Three options, two picked (Spinner) - 9

Four options, two picked - 16

Six options, two picked - 36

So for options, if we pick two with repetition, we get permutations.

Try the discovery worksheet about permutations!

An Introduction to Permutations

Permutations are combinations where order matters. (These are like the ones we did today.)To calculate permutations with repetition where order matters for possible values and choosing of them, we get…

𝑛𝑟

So if we rolled a 10 sided die four times, how many permutations exist?

104¿10000