–
P(A and B) = P(A) x P(B) e.g. rolling 2 sixes = P(6 and 6)=1/6 x 1/6
▪
Two events are dependent when the outcome of the first event
influences the outcome of the second event. The probability of two
dependent events is the product of the probability of A and the
probability of B AFTER A occurs.
–
P(A and B) = P(A) x P(B after A)
–
eg. Select 2 red cards from a playing deck.
P(first red) = 26/52 and selecting a
second red after the first is 25/51 (remember we removed one red card from the
pack), therefore P(red and red) = 26/52 x 25/51

42
Basic maths of probabilities (mutually exclusive)
▪
Two events are mutually exclusive when two events cannot happen at the
same time. The probability that one of the mutually exclusive events occur is
the sum of their individual probabilities.
–
P(A or B) = P(A) + P(B) eg. Rolling a 5 or 6 on a single toss of die = 1/6 +1/6 = 2/6
▪
Inclusive events are events that can happen at the same time. To find the
probability of an inclusive event we first add the probabilities of the individual
events and then subtract the probability of the two events happening at the
same time.
–
P(A or B) = P(A) + P(B) – P(A and B)
▪
E.g. What is the probability of drawing a black card or a ten in a deck of
cards?
–
There are 4 tens in a deck of cards P(10) = 4/52
–
There are 26 black cards P(black) = 26/52
–
There are 2 black tens P(black and 10) = 2/52
–
P(10 or black) = 4/52 + 26/52 – 2/52
▪
If scenarios are mutually exclusive and exhaustive (ie. Fully describe all
events and consequences) then the scenario probabilities must sum to one.

43
Is Scenario 2 of the Coin Toss (Lecture 1) really
lower risk?
Scenario 1
- worst case loss is 50% chance of losing
$1,000.
Scenario 2
- we need to model the outcome of all possible
combinations of winning and losing over 1,000 trials.
What is the likelihood of losing $1000?
What is the likelihood of losing more than $50
Because we know the statistical process describing the coin
toss (eg. 50/50 chance of head or tail) we can use theory to
answer these questions.

44
Coin toss payoff after 4 tosses (trials)
Tail =-$1
4T = -$4
4H = $4
3H+1T=$2
3T+1H=-$2
2H+2T=$0
Head =$1
After 4 trials worst case
loss = -$4
Likelihood is 0.5 x 0.5 x
0.5 x 0.5 =0.5
4
=
0.0625
Worst case after 1,000
trials = $1,000 loss with
likelihood =
0.5
1,000
=
9.332636e-302 !!!
Almost
impossible
to lose $1,000
But not
impossible

45
Using computer simulation we can estimate low
probability events
▪
What do you think is the probability of losing more than
$100? What about $50?
–
Run 1,000 consecutive trials of coin tosses
–
Repeat exercise 10,000 times
–
Count how often lose >$100
9/10,000=0.09%
482/10,000=
4.82%

46
Advantages of incorporating probabilities in
likelihood assessment
▪
Estimates of probabilities are typically used to supplement
qualitative descriptions
▪
Usually described as a range over a specified time period
Scale
Description
An event has the probability of occurring within
the next 10 years with probability in the range of:
1
Very unlikely
Less than 20% (but greater than 0%)
2
Unlikely
Between 21% and 40%
3
Somewhat likely
Between 41% and 60%
4
Likely
Between 61% and 80%
5
Very likely
More than 80% (but less than 100%)

Decision-making under conditions of
uncertainty and risk aversion

48
Decision Tree analysis – making decisions under
uncertainty
▪

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