Question

A Box has 1024 fair and 1 two-headed coin. You pick a coin randomly, make 10 flips and get all H.

(a) You flip the same coin you picked 100 times. What is the expected number of H?

(b) You flip the same coin you picked until you get H. What is the expected number of flips you make?

Answer 1

Hello,

Solution to this problem is totally depends upon the chosen coin (i.e. either it is fair or two-headed coin). Hence we have to find firstly below two, before finding solution to question part (a) and (b).

1. Probability of coin being fair coin is equal to P (F | Flips)

2. Probability of coin being Two Headed (Biased) coin is equal to P ( B | Flips)

Now, using Bayes' formula to calculate above two.

1.

P (F | Flips) = [ P(Flips | F) * P(F) ] / P(Flips)

since this is fair coin, hence probability of getting head in each flip is 0.5 (i.e. P(Flips | F))

P(F) is the probability of getting fair coin from all the available coins (i.e. 1024/1025)

Hence,

P(F | Flips) = [ (0.5)^10 * (1024/1025) ] / [ [ (0.5)^10 * (1024/1025) ] + [ (1)^10 * (1/1025) ] ]

= (1 / 1025) / ( 2 / 1025)

= 1/2

= 0.5

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2.

P (TH | Flips) = [ P(Flips | TH) * P(TH) ] / P(Flips)

since this is two headed coin, hence probability of getting head in each flip is 1 (i.e. P(Flips | TH))

P(TH) is the probability of getting two headed coin from all the available coins (i.e. 1/1025)

Hence,

P(TH | Flips) = [ (1)^10 * (1/1025) ] / [ [ (0.5)^10 * (1024/1025) ] + [ (1)^10 * (1/1025) ] ]

= (1 / 1025) / ( 2 / 1025)

= 1/2

= 0.5

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Question part (a):

Now, we are going to find out the probability of getting head in the next flip firstly:

which is equal to

P( H ) = P( Head when fair coin ) *  P( F | Flips ) + P( Head when Two Headed coin ) *  P( TH | Flips )

P( H ) = ( 0.5 * 0.5 ) + ( 1 * 0.5 )

P( H ) = 0.75

Now, according to question part (a) , we have to calculate expected occurrence of head when flipping the same coin 100 times,

Expected number of heads in 100 flips = P ( H ) * 100

= 0.75 * 100

= 75

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Question Part (b) :

Now, we have to find expected number of flip we have to make for getting a head, and it is calculated as : ( 1 / P( H ) )

= ( 1 / 0.75 )

= 1.33

By rounding it off to upper side it is 2.

So minimum of 2 attempts or flips required to get a head.

That's all!!

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Hope you will understand it well now and in case of any doubt, please feel free to ask them. I will be happy to answer them.