Transcript Introduction to Probability

Introduction to Probability
What is it?
• For a random phenomenon, individual
outcomes are not certain, but there is a
regular distribution of outcomes in the long
run
• The probability of an outcome is its longterm relative frequency.
Probability vs. inference
• If a coin is tossed 100 times, you would
expect to see 50 heads
• However, there is some (non-zero)
probability that it only comes up heads 25
times
• If there were 100 flips, and it only came up
heads 25 times, you might infer that it
wasn’t a fair coin.
What are the possible outcomes?
• Want to make a list of possible outcomes,
then find probability for each outcome
• Sample space is the set of all possible
outcomes
• Events are specific outcomes or set of
outcomes in the sample space
Probability Scale
You can draw a scale from Impossible to Certain:
Estimated Probability
Certain
Likely
Even Chance
Unlikely
Impossible
What is the probability of scoring less than 5 when rolling
a fair 6-sided die?
Introduction to Probability
•Rather than using a descriptive scale, we use a
numerical scale
•Probability is a numerical measure of the
likelihood that an event will occur.
• Probability values are always assigned on a
scale from 0 to 1.
• A probability near 0 indicates an event is very
unlikely to occur.
• A probability near 1 indicates an event is
almost certain to occur.
Useful facts about probability
• Probability cannot be less than 0 or greater
than 1.
• All possible outcomes together must have
probability 1.
• Probability of an event occurring is 1 minus
probability that it does not occur.
• If 2 events have no outcomes in common,
probability that one or the other occurs is
sum of their individual probabilities
Example: You roll a six-sided die whose sides are
numbered from 1 through 6. Find the probability of
a. rolling a 4
=
number of ways to roll a 4
number of ways to roll the die
= 1
6
b. rolling an odd number
= number of ways to roll an odd number
number of ways to roll the die
= 3
6
= 1
2
c. rolling a number less than 7
= number of ways to roll less than 7
number of ways to roll the die
= 6
6
= 1
Combining Probabilities
What is the probability of scoring either a 2 or
a 6 from a single roll of a fair die?
Number of ways an event can happen
Total number of possible outcomes
= 26
= 13
1
1
And
p(5)
=
Or p(1) = 6
6
2 = 1
1
+
1
=
=
6
3
6
So p(1 or 5) =
1
1
+
6
6
Combining probabilities
• What happens if we need to calculate the
probability of one event occurring and
another event occurring?
• What is the probability of rolling a 4 with a
die and tossing a head with a coin?
• How many outcomes are there?
Combining probabilities
1
2
3
4
5
6
H
H1
H2
H3
H4
H5
H6
T
T1
T2
T3
T4
T5
T6
12 possible outcomes.
Only one is H plus a 4
So p(4 and head) = 1
12
Or: The probability of rolling a 4 is 1
6
The probability of throwing a head is 1
2
1  1 = 1
1
1
of
So p(4 and head) =
of the time
2
12
6
6
2
Probability
• Throwing dice, tossing coins, picking cards
- in each case each outcome is equally
likely
• It is not always the case that all possible
outcomes of a trial are equally likely.
• Sometimes we cannot calculate the
probability, but can only estimate it
Probability
• Will it rain tomorrow? Will Team A win
over Team B?
• Each outcome is not equally likely.
• For one team playing a game against
another, there are three possible outcomes:
Win, Lose, Draw
• Probability of winning is not one third.
Probability
• The probability that they will win can only
be estimated, using other information,
mainly past performance of the two teams.