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
Flipping a Coin…Twice!
Let’s consider flipping an unbiased coin twice.
What are the possibilities we can get?
Probability (of happening)
Let’s list them…
P(H, H) = 1 4
① Heads, Heads
P(H, T) = 1 4 There are 4
② Heads, Tails
P(T, H) = 1 4 combinations.
③ Tails, Heads
P(T, T) = 1 4
④ Tails, Tails
Note – Heads, Tails is
Why is each ¼?
different to Tails, Heads.
1
# of successes
When listing possible
outcomes, order does
# of outcomes
4
have meaning.
=
Flipping a Coin…Twice!
How about the following?
Order isn’t mentioned.
P(A Heads and a Tails) = 2 P(H and T) = P(H, T) + P(T, H).
4
We always get heads or
P(A Heads or a Tails) = 1
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
No. Both events are
independent with the same
probabilities for each outcome.
P(Tails) = ½
Yes, their outcomes
don’t affect each other.
1, 2, 3
1
P(2) =
3
P(H, T) = ¼
P(At least one heads) =
P(Exactly one heads) =
P(No tails) =
1
4
2
4
3
4
1, 1 2, 1 3, 1
1, 2 2, 2 3, 2
1, 3 2, 3 3, 3
1
P(1, 3) =
9
1
9
P(2, 2) =
P(No 4) = 1
Answers - Hard
P(Both Even) =
16
P(2, 4) =
9
36
P(Both less than or equal to 4) =
1
16
12
2
16 6
P(A 1 and a 3) =
P(Exactly one 2) =
P(Total of 7) =
2
16
P(Total of less than 3) =
2
P(A 3 and a 4) =
36
6
P(Total of 7) =
36
16
36
16
1
16
P(H, 4) =
1
12
P(Tails and greater than 2) =
4
12
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 𝑥 with repetition
allowed (picking the same number twice).
Try the discovery worksheet about permutations!
Two options, two picked (Coin) - 4
Three options, two picked (Spinner) - 9
Four options, two picked Six options, two picked -
16
36
So for 𝑛 options, if we pick two with repetition, we
get 𝑛2 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 𝑛 possible values and choosing 𝑟
of them, we get…
𝑛
𝑟
So if we rolled a 10 sided die four times, how many
permutations exist?
4
10
= 10000