Probability and Expected Value

Lecture 4

Standard notation for expressing the probability of an outcome is…

p(outcome) = ____ ← (the blank is a decimal between 0.0 → 1.0)

Formula for probability is…

p(outcome) = (number of independent results that would qualify as outcome) / (number of possible outcomes for the event)

Ex. p(random item from my pocket is a phone) = ¼ (4 being phone or kets or mint or chapsticks) = 0.25

CONVERTING PROBABILITY INTO PERCENTAGE…

MULTIPLY BY 100: 0.25 → 0.25×100 → 25% chance 

Mutually exclusive 

Two outcomes are mutually exclusive if it would be impossible for both outcome to happen 
For example 

• A coin can’t land on both head and tails

• A football team can’t both win and lose 

OR (add) vs AND (multiply)

When calculating the probability of MUTUALLY EXCLUSIVE outcome1 OR outcome2, ADD (+)…

Example: What’s the prpbabiltiy of rolling 2 OR 4 OR 6?

• p(2 or 4 or 6) = 1/6 + 1/6 + 1/6 = 3/6 = 0.5 

For p of MUTUALLY EXCLUSIVE outcome1 AND THEN ALSO outcome2, MULTIPLY (*)….

Example: What’s the probability of rolling 2 AND THEN 4 AND THEN 6?

• p(2, and then 4, and then 6) = p(2) x p(4)  x p(6) = 1/6 × 1/6 × 1/6 = 1/216 

Gamblers Fallacy

Our tendency to assume a certain independent outcome is “due”, because this outcome has not occurred for several prior identical independent event 

Hot-Hand Fallacy

Tendency to assume that a certain independent outcome is “on a roll”, because it was the result of several prior independent events

Expected Value

If you could make the bet 10-trillion times, what would be the avg outcome?

EV= [value of outcome1] x [p(outcome1)] + [value of outcome2] x [p(outcome2)] 
(same formula written more concisely) EV = o₁ x p₁ + o₂ + p<sub>2

Examples 
Flip a coin. If it’s heads, you win $2. If it’s tails, you lose $1 (WHEN LOSE DON”T FORGET NEGATIVE)
EV = [$2 × 0.5] + [$1 x -.5] = $1 + [-$0.50] = $0.50</sub> 

Total Possible Outcome

= number of outcomes of each event multiply by the other event