Question

You have a biased coin for which p(h)=pp(h)=p. you toss the coin 2020 times. what is the probability that you observe 88 heads and 1212 tails; you observe more than 88 heads and more than 88 tails?

Answer 1

The question describes a binomial probability with p(h) = p, then p(t) = 1 - p and number of trials (n) = 20

The probability of a binomial distribution is given by


`P(x)=\, ^nC_xp^x(1-p)^(n-x)`

Part A:

The probability of observing 8 heads and 12 tails is given by:


`P(8)=\, ^(20)C_8p^8(1-p)^(20-8)=\, ^(20)C_8p^8(1-p)^(12)`



Part B:

You observe more than 8 heads and more than 8 tails, when you observe 9 heads and 11 tails, 10 heads and 10 tails, and 11 heads and 9 tails.

Therefore, the probability of observing more than 8 heads and more than 8 tails is given by:


`P(9)+P(10)+P(11) \\ \\ =\, ^(20)C_9p^9(1-p)^(20-9)+\, ^(20)C_(10)p^(10)(1-p)^(20-10)+\, ^(20)C_(11)p^(11)(1-p)^(20-11) \\ \\ =\, ^(20)C_9p^9(1-p)^(11)+\, ^(20)C_(10)p^(10)(1-p)^(10)+\, ^(20)C_(11)p^(11)(1-p)^(9)`