Flipping coins by Harvard Fat Chance

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Flipping coins

video links

Bilibili p31-33

my note videos p30-31

key question

How probability come down to counting

Problem

6 coin flipping, if the outcome is 3 heads 3 tails, win 20 dollars; otherwise, lose 10 dollars; will you bet?

outcome

• 

Assumptions

• fair coin = p = 0.5 = 50%
• independence: first H or T, no effect on second H or T
• so, all outcomes are equally likely

two questions

• how many possible outcomes are there
• how many involve 3 heads and 3 tails

solution to question 1

• steps = 6 coins flipping
• options = 2 head or tail
• independent, repetition allowed = $2^6$

solution to question 2

• steps = take 3 coins to be heads = k
• options = 6 positions or flips = n
• inside 3-heads group, the order no matter = collection count
• $ = (^n_k)= (^6_3)$

favorable outcome

• 3 heads and 3 tails

probability formula

• 
• p = $(^6_3)/2^6$

Practice

8 coin flipping, what is the probability of getting 4 heads?

Experiment

• steps = 8
• options = 2
• constraint = 4 heads out of 8 flips

total

• 8 steps, each step has 2 options, repetition allowed = $2^8$

interest

• 4 heads(4 tails), not 1 head, or 2, 3, 5, 6, 7, 8 heads
• 8 slots, choose 4 slots as a group, uniqueness defined by its location distinction
• order inside the 4 slots does not matter
• $(^8_4)$

probability = $(^8_4)/2^8$

What is the probability of getting 4, 5, or 6 heads out of 10 coin flipping?

• 4 heads prob = $(^{10}_4)/2^{10}$
• 5 heads prob = $(^{10}_5)/2^{10}$
• 6 heads prob = $(^{10}_6)/2^{10}$
• add them together