Both questions below are on the law of averages. (a) The results of 50 spins of

Question

Both questions below are on the law of averages. (a) The results of 50 spins of a penny are shown in the figure below.

(i) Was the first spin heads or tails? the second spin? the third? the 50th?

(ii) Use these data to estimate the probability that this penny will land heads when it is tossed. (Ballpark answer is fine)

(b) Suppose a fair six sided die will be rolled some number of times.

There are two choices: (a) 12 rolls and you win a dollar if an even number is rolled 3 or more times.

(b) 36 rolls and you win a dollar if an even number is rolled 9 or more times.

Explanation / Answer

a)

i) From the Spin Number vs % of heads chart, it is clearly evident that the % of heads is at 0% for spin numbers 1,2 and goes to 33% for spin number 3. Also, the % of heads has dropped after the 50th spin.

This shows the first, second, third spins are Tails, Tails, Heads respectively & the 50th spin is Tails

ii) As the number of tosses increases, the % of heads has stayed flat and closer to 42.5% at 50 trials. So, going by the law of averages, heads seems to be catching up and this doesn't look like a biased coin and will end up giving heads with a probability of 0.5

b) a) No. of rolls N=12. p = Prob(Even) = 3/6 = 0.5; q = Prob(Odd) = 0.5

P(Even=x) is given by NCR (p^R)*(q^(N-R))

P(Even >=3) = 1 - P(Even<3) = 1 - [P(Even)=0+P(Even)=1+P(Even)=2]

= 1 - [0.01928] = 0.9807

b) No. of rolls N=36.

P(Even>=9) = 1 - P(Even<9)

= 1 - [P(Even)=0+P(Even)=1+P(Even)=2+P(Even)=3+P(Even)=4+P(Even)=5+P(Even)=6+P(Even)=7+P(Even)=8]

= 1 - 0.000596 = 0.999403

The probability if winning a dollar is higher if the choice made is to roll 36 times to get 9 or more evens than to roll 12 times to get 3 or more evens.

This has been shown by computing probabilities from bernoulli trials but can also be intuitively picked going by the law of averages!

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