– 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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