Question

Question 12 suppose i have either a fair coin or a bent coin, and i don't know which. the bent coin has a 60% probability of coming up heads. i throw the coin ten times and it comes up heads 8 times. what is the probability i have the fair coin vs. the probability i have the bent coin? assume at the outset there is an equal (.5,.5) prior probability of either coin. *please note that in order to fit the entire formula in the feedback, probability has been abbreviated to "prob."

Answer 1

The likelihood is that probability as a function of the unobservable parameter p with the data held fixed (i.e. the number of successes in 10 trials), thus it isp↦10C8p^8(1−p)^2and in this case thegiven value of p, is  0.5, 0.5  Thus you could actually represent the likelihood as a pair:10C8(0.5)^8(0.5)^2=0.043945

Answer 2

Given : i have either a fair coin or a bent coin, and i don't know which. the bent coin has a 60% probability of coming up heads.   i throw the coin ten times and it comes up heads 8 times.

To find : probability i have the fair coin vs. the probability i have the bent coin

Solution:

Case 1 : Fair coin

Head Probability p = 1/2

Tail Probability q = 1/2

n = 10

Comes Head 8 times

Probability = ¹⁰C₈(1/2)⁸(1/2)²

= 45 / 2¹⁰

= 45/ 1024

= 0.044

Case 2 : Bent coin

Head Probability p = 0.6 = 3/5

Tail Probability q = 2/5

n = 10

Comes Head 8 times

Probability = ¹⁰C₈(3/5)⁸(2/5)²

= 45 *3⁸ * 2²/ 5¹⁰

= 45  * 26244/97,65,625

= 0.121

Probability of bent coin = (0.121)/(0.121 + 0.044)  = 0.733

Probability of fair coin = (0.044)/(0.121 + 0.044)  = 0.267

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