Expected Value Strategizing based on Harvard Fat Chance

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Expected Value Strategizing

key questions

Expectation can help decisions but with many constraints

how to compare many games for the best one using Expectation?

Video links

original course videos p49-51

my note videos p48-49

problem

What is the expected value for this experiment above?

one dice

• total = 6
• one 6 and no 1s = dice1(1)
• target prob = 1/6

two dices

• total = dice1(6) x dice2(6)
• no 1s = dice1(5) x dice2(5) = A = 25
• one or more 6s and no 1s = A - no 6s and no 1s = A - dice1(4) x dice2(4) = 25-16 = 9

three dices

• total = dice1(6) x dice2(6) x dice3(6) [order matters here, collection is not required]
• no 1s = dice1(5) x dice2(5) x dice3(5) [order matters here, collection is not required] = A = 125
• one or more 6s and no 1s = A - no 6s and no 1s= A - dice1(4) x dice2(4) x dice3(4) = target count = 125-64 = 61
• other = total - target count

Practice

problem

roll n dice, desired outcome = get a 5 or 6 but no 1s, what is the probability?

total = dice1(6 options) x dice2(6) x ... diceN = $6^n$

target on no 1s = dice1(5 options) x dice2(5 options) x ... diceN = $5^n$ = A

target on get a 5 or 6 and no 1s = A - no 5 nor 6 and no 1s = A - dice1(6-3) x dice(3) x ... x diceN(3) = A - $3^n$

target probability = $\frac{5^n - 3^n}{6^n}$